The solutions are 150, 2.5, 39 and 101. The solutions have been obtained using the arithmetic operations.
What are arithmetic operations?
In all, there are four basic mathematical operations which are applicable for all the real number. The operations are as follows:
1.Addition(‘+’) operation which gives the sum of the numbers.
2.Subtraction(‘-’) operation which gives the difference of the numbers.
3.Multiplication(‘×’) operation which gives the product of the numbers.
4.Division(‘÷’) operation which gives the quotient of the numbers.
1. The sum of 6 and the square of 12
The square of 12 = 12*12 = 144
Sum of 6 and 144 = 6 + 144 = 150
2. The quotient of 30 and the sum of 5 and 7
The sum of 5 and 7 = 5 + 7 = 12
The quotient of 30 and 12 = 30 ÷ 12 = 2.5
3. The difference of 4 cubed and 25
The cube of 4 = 4*4*4 = 64
The difference of 64 and 25 = 64 - 25 = 39
4. 5 more than the product of 12 and 8
The product of 12 and 8 = 12*8 = 96
5 more than 96 = 5 + 96 = 101
Hence, the solutions are 150, 2.5, 39 and 101.
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let abcd be a regular tetrahedron and m,n distinct points in the planes abc and adc respectively. show that the segments mn,bn,md are the sides of a triangle.
The segments MN, BN, and MD are the sides of a triangle.
Let AM intersect BC at point P and let AN intersect CD at point Q. Spin A about PQ to obtain E where AE = AB. We want to prove that EMN is the required triangle. For that, it is adequate to show that EM = MD and EN = BN.
Let h be the distance of A from the BCD. Then the distance of A from the line PQ is minimum h, so point E exists provided that the side length maximum is 2h. It is √(3/2) h.
We have by construction AD = AE and EP = AP. Also, since P lies on the line BC, then PA = PD. Hence PA = PA, PD = PE, AD = AE, so PAD and PAE are congruent triangles. Hence E can also be acquired by spinning D about AP. Since M lies on AP, it gives EM = DM.
Similarly, with B and D interchanged and M and N interchanged, we get EN = BN.
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Sample Questions Chapters 3, 12, 13, 14. All of the following contribute to the success of incumbent members of Congress n election campaigns EXCEPT Incumbents.
All of the following contribute to the success of incumbent members of Congress n election campaigns EXCEPT Incumbents tend to understand national issues better than do their challengers.
What does the term incumbent mean?The term "incumbent" has several meanings depending on the context, but it generally refers to someone who holds a particular position or office. The most common usage of this term is in the context of politics, where an incumbent is the current holder of a political office, such as a president, governor, or member of Congress. An incumbent candidate is the candidate who is currently holding a political office and is running for re-election.
In other fields, the term incumbent is used to describe someone who holds a particular job, position or role and is responsible for certain tasks or duties, for example an incumbent employee, an incumbent contractor, an incumbent provider, etc.
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Complete question not found
Business XYZ has the following projected budget
and actual budget for a given month:
Budget Actual
Payroll $24,000 25,275
Inventory 12,700 12,100
Rent
7,500 7,700
Services 5,000 5,100
Calculate the percent change that the total of all
categories is over budget.
[?] %
Give your answer as a percent rounded to the nearest tenth.
Enter
PH
Skip
Answer:
The total budget for all categories is $49,200 and the total actual is $50,175, for a percent change of 1.8%. Rounded to the nearest tenth, the percent change is 1.9%.
give me brainiest
Which expression is a factor of -2x2 - 11x - 20
No real Factors
(2x + 10)
(x - 10)
(2x - 5)
The given expression will not have real factors. The correct option is A.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that the quadratic expression is -2x² - 11x - 20. The expression will be solved as below:-
E = -2x² - 11x - 20
Apply the quadratic factorization formula,
x = [ -b ± √( b² - 4ac ) ] / 2a
x = [ 11 ± √ ( 121 - ( 4 x - 2 x - 20 ) ] / ( 2 x - 2 )
x = [ 11 ± 39i } / -4
The roots of the quadratic equation are not real.
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Answer:
no real factors
Step-by-step explanation:
i took the test
Macmillan Learning
Assume the Pacific Ocean has a surface area of 1.5 x 108 km² and an average depth of 3700 m.
Calculate the volume of water in the ocean in liters:
5.6 x 10¹2 L
5.6 x 10¹7 L
O 5.6 x 10¹¹ L
5.6 x 10^20 L
The volume of water in the ocean in litters, 5.6 x 10¹¹ L.
How do you calculate the size of the Pacific Ocean?In reality, the area of one side multiplied by the depth or height of the design yields the volume of cylinders and prisms. The area of the final shape, or the height and depth of the prism or cylinder, is the basic formula for prisms' and cylinders' volumes.
The area of the final shape, or the height and depth of the prism or cylinder, is the basic formula for prisms' and cylinders' volumes.
In reality, the area of one side multiplied by the depth or height of the design yields the volume of cylinders and prisms.
Volume is calculated as Area x Depth.
1.5 × 108/3700 = 5.6 x 10¹¹ L.
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Construct a 3x3 matrix A, with nonzero entries, and a vector b in R3 such that b is not in the set spanned by the columns of A. Choose the correct answer below : A. A= | 1 1 1 | | 3 | B. | 1 1 1 | |4| . |2 2 2 | b = | 6 | A= | 2 2 2 | b = |5| . |3 3 3 | | 9 | | 3 3 3 | |6| . . C. A= | 1 1 1 | | 1 | D. | 1 2 3 | |3| . |2 2 2 | b = | 2 | A= | 2 1 1 | b = |2| . |3 4 5 | | 5 | | 3 3 2 | |1|
A 3x3 matrix A, with nonzero entries, and a vector b in R3 such that b is not in the set spanned by the of A= | 1 1 1 | | 1 | . |2 2 2 | b = | 2 |
In order for b to not be in the set spanned by the columns of A, we need to ensure that the system of equations Ax=b has no solution. This means that the system of equations needs to be inconsistent. We can achieve this by making sure that the determinant of A is not equal to zero, i.e. the matrix must be invertible.
The matrix given in option C satisfies this requirement. A = | 1 1 1 | | 2 2 2 | | 3 3 3 |
The corresponding vector b is | 1 | | 2 | | 3 |
Thus, the correct answer is C.
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Kathleen has 21 marbles some of the marbles are green the other 12 marbles are blue How many green marbles does she have?
Answer: 9
Step-by-step explanation:
21 - 12 = 9
Peyton had a balance of -$21.46 in his bank account according to his
last statement. He immediately deposited $35.50 into his account.
What was Peyton's account balance AFTER he made the deposit?
Answer: $14.04
Step-by-step explanation: -21.46+35.50 = 14.04
the quotient of ten less than negative four and negative one half
Answer:
Step-by-step explanation:
28
estimate the distance (in mi) the race car traveled during this time period using the velocities at the beginning of the time intervals
The distance the race car traveled during this time period using the velocities at the beginning of the time intervals is 23.886 mi
Use areas of rectangles to approximate the area under the curve. The width of each rectangle is 10 sec = 1/36 hrs and the lengths are the velocities.
LH = 1/36 ( 183.9 + 167 + 107.6 + 98.8 + 125.5 + 177.1)
LH=1/36(862.9)
LH≈ 23.886 mi
To find the right-hand approximation:
RH = 1/36 (167 + 107.6 + 98.8 +125.5 +177.1 + 176.6)
RH=1/36(852)
RH≈ 23.683 mi
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all of the following are possible steps of scientific investigation except for . group of answer choices the development of one or more working hypotheses or models to explain facts development of observations and experiments to test the hypotheses assumption of conclusions without prior experimentation or observation the collection of scientific facts through observation and measurement
All of the following are possible steps of scientific investigation except for
C. assumption of conclusions without prior experimentation or observation
All of the following are possible steps of scientific investigation except for.
A. the development of one or more working hypotheses or models to explain facts
B. development of observations and experiments to test the hypotheses
C. assumption of conclusions without prior experimentation or observation
D. the collection of scientific facts through observation and measurement
All of the following are possible steps of scientific investigation except for
C. assumption of conclusions without prior experimentation or observation
At the core of biology and other sciences lies a problem-solving approach called the scientific method. The scientific method has five basic steps, plus one feedback step:
Make an observation.
Ask a question.
Form a hypothesis, or testable explanation.
Make a prediction based on the hypothesis.
Test the prediction.
Iterate: use the results to make new hypotheses or predictions.
The scientific method is used in all sciences—including chemistry, physics, geology, and psychology. The scientists in these fields ask different questions and perform different tests. However, they use the same core approach to find answers that are logical and supported by evidence.
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Captain Ben has a ship, the H.M.S Crimson Lynx. The ship is five furlongs from the dread pirate Luis and his merciless band of thieves.
If his ship hasn't already been hit, Captain Ben has probability
2
3
3
2
start fraction, 2, divided by, 3, end fraction of hitting the pirate ship. If his ship has been hit, Captain Ben will always miss.
If his ship hasn't already been hit, dread pirate Luis has probability
1
3
3
1
start fraction, 1, divided by, 3, end fraction of hitting the Captain's ship. If his ship has been hit, dread pirate Luis will always miss.
If the Captain and the pirate each shoot once, and the pirate shoots first, what is the probability that the pirate hits the Captain's ship, but the Captain misses?
The probability that the pirate hits the Captain's ship, but the Captain misses is 1/6.
What is probability?It is the chance of an event to occur from a total number of outcomes.
The formula for probability is given as:
Probability = Number of required events / Total number of outcomes.
We have,
P(Captain Ben hit the ship) = 2/3
P(Captain Ben misses to hit the ship) = (1 - 2/3) = 1/3
P(Pirate hit the ship) = 1/3
P(Pirate misses to hit the ship) = (1 - 1/3) = 2/3
Now,
The probability that the pirate hits the Captain's ship, but the Captain misses.
= P(Pirate hit the ship) x P(Captain Ben misses to hit the ship)
= 1/3 x 1/3
= 1/6
Thus,
The probability is 1/6.
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Show me 2/4 comparing 5/8 on a number line
Using a number line, the number 5/8 is greater than 2/4
How to compare the numbers on a number lineFrom the question, we have the following parameters that can be used in our computation:
Comparing 2/4 and 5/8 on a number line
Here is a number line that shows the comparison of 2/4 and 5/8:
1/2 2/4 3/4 5/8 3/4 2/4 1/2
--|---------|---------|---------|---------|---------|--
As you can see, 2/4 is located halfway between 1/2 and 3/4, while 5/8 is located halfway between 3/4 and 3/4.
Therefore, 5/8 is larger than 2/4.
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the question is in the picture
The positive interval and the negative interval for the function given is (-∞, 3) and (3, ∞).
What is Function?A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.
Given function is f(x) = [tex]-2\sqrt{x+1} +4[/tex]
Positive interval of the function is the values of x when function > 0.
To get the function > 0, the value [tex]2\sqrt{x+1}[/tex] < 4.
[tex]\sqrt{x+1}[/tex] < 2
x + 1 < 2²
x + 1 < 4
x < 3
Positive interval is (-∞, 3).
Negative interval of the function is the values of x when function < 0.
To get the function value < 0
[tex]-2\sqrt{x+1} +4[/tex] < 0
[tex]2\sqrt{x+1}[/tex] > 4
[tex]\sqrt{x+1}[/tex] > 2
x + 1 > 2²
x + 1 > 4
x > 3
Negative interval is (3, ∞).
Hence the positive interval of the function is (-∞, 3) and the negative interval is (3, ∞).
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steven is cutting a board to use for a project.
the board is 10 feet long. he first has to cut off 1 1/2 feet from one end of the board. he has to cut the remaining section board into 3 pieces.
what is the length of each equal piece?
write and solve an equation to represent this situation. write answer as a mixed number
The length of each equal piece is 17/6 feet, the mixed fraction is 2 5/6.
What are proper and improper fractions and how to convert mixed fraction to simple fraction?There is a fraction, containing numerator(upper value) and denominator(lower value).
When the numerator is less than the denominator, the fraction is called proper fraction, otherwise it is called improper fraction.
A proper fraction is also called fraction which is less than 1.
And an improper fraction is ≥ 1
A mixed fraction contains a sum of whole number and a proper fraction.
Given;
The board is 10 feet long
First he has to cut off 1 1/2 feet from one end of the board
Now, after the first cut remaining board will be;
=10-1 1/2=17/2
After cutting it in 3 equal pieces
=17/2/3
=17/6
=2 5/6
Therefore, the answer in mixed fraction will be 2 5/6
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to find the average amount of money that american college students spend on food per week, a researcher surveys 300 college students across the u.s.
The result of the calculation was $93.41 per week.
To find the average amount of money spent on food by American college students per week, a researcher conducted a survey of 300 college students across the U.S. The researcher collected data on the amount of money spent by each student on food per week and then calculated the average using the formula: Average = (sum of all the scores)/(number of scores). The researcher then added up the money spent by all 300 students and divided it by 300 to get the average amount of money spent on food per week by American college students. The result of the calculation was $93.41 per week.
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6. How can you be certain that the angles you have created form a square inside the circle? (4
points for a thorough explanation)
To be certain that the angles created form a square inside a circle, it is important to first understand the properties of a square and a circle.
How to sure that Angles created form a Square inside a Circle?A square is a four-sided polygon with all sides of equal length and all angles of 90 degrees. This means that if we draw a square inside a circle, the four corners of the square should be tangent to the circle.
A circle is a shape with all points on the circumference equidistant from the center. This means that if we draw a square inside a circle, the distance from the center of the circle to the midpoint of each side of the square should be equal to the radius of the circle.
By verifying that the above two conditions are met, we can be certain that the angles created form a square inside the circle. Additionally, we can also check that all the four angle of the square is 90 degrees and the opposite sides are parallel.
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The second bus company that Ms. Baudy contacted wanted to charge each student $5.00 for
roundtrip service. If the total price of the trip per student was $7.00, approximately what
percentage of the trip would go towards the price of the bus?
If the total price of the trip per student was $7.00, approximately the percentage of the trip that would go towards the price of the bus is 71.43%.
What is the percentage?The percentage is the ratio of a variable or value to another.
Percentages are fractional values, computed as the quotient of a numerator over a denominator multiplied by 100.
The bus roundtrip charge per student = $5.00
The trip price per student = $7.00
Percentage of the trip going towards the price of the bus = 71.43% ($5/$7 x 100).
Thus, we expect that 71.43% of the trip amount will be consumed for the bus charter.
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At a family reunion, each of Shamar's aunts and uncles is getting photographed once. The aunts are taking
pictures in groups of 5 and the uncles are taking pictures in groups of 10. If Shamar has the same total number
of aunts and uncles, what is the minimum number of aunts that Shamar must have?
The minimum number of aunts that Shamar must have based on the information provided is 10.
What are the possible number of uncles and aunts?It is known that uncles will take a photograph in groups of 10, while aunts will take a photograph in groups of 5. Based on this, the possible number for each are:
Uncles: 10, 20, 30, 40, 50
Aunts: 5, 10, 15, 20, 25
How to find the minimum number of relatives?In this case, the minimum number is the LCM or Least Common Multiple and based on the previous list, this number is 10.
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Average monthly expenditure (£)
600
500
400
300
200
100
Clip 15
Key:
2010
2020
I
Transport Utility Food Housing Leisure
bills Category
a) Which category shows a decrease
in the average monthly expenditure
between 2010 and 2020?
leisure
b) How much more was the average
monthly expenditure on Food in
2020 than in 2010?
c) The total average monthly
expenditure in 2010 was £1200.
What was it in 2020?
Help on question c
Answer:
the total average monthly expenditure in 2020 was £2400.
Step-by-step explanation:
c) To find the total average monthly expenditure in 2020, we need to add up all the individual expenditures in each category. From the given data, we have:
Average monthly expenditure in 2020 = 600 + 500 + 400 + 300 + 200 + 100 = £2400
So, the total average monthly expenditure in 2020 was £2400.
The number of tricycles in the preschool play yard was?
2
4
No answer text provided
3
The number of tricycles in the preschool play yard is given as follows:
10.
How to obtain the number of tricycles?The variables for the system of equations are given as follows:
Variable x: number of bicycles.Variable y: number of tricycles.There are 30 seats, hence:
x + y = 30
x = 30 - y.
There are 70 wheels, hence:
2x + 3y = 70.
Replacing x = 30 - y on the second equation, the number of tricycles is obtained as follows:
2(30 - y) + 3y = 70.
y = 10.
Missing InformationA preschool playground has both bicycles and tricycles. There is a total of 30 seats and 70 wheels. What is the number of tricycles?
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20 people fit comfortably in a 10 feet by 10 feet area. Use this value to estimate the size of a
crowd that is 7 feet deep on ONE side of the street along a 1 mile section of a parade route.
The size of the crowd is 7392 people.
What is ratio?The ratio is a numerical relationship between two values that demonstrates how frequently one value contains or is contained within another.
20 people fit comfortably in a 10 feet by 10 feet area.
So,
the ratio of the number of people per unit area = 20/(10 × 10) = 20/100 = 1/5
The area, A, occupied by the crowd = 7 feet × 1 mile × 1
1 mile = 5280 feet
A = 7 feet × 5280 feet × 1 = 36960 feet².
Now, number of people = 36960 x 1/5
= 7392 people.
Therefore, the required value is 7392 people.
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Joe’s garden hose fills the pool in hours. His neighbor has a thinner hose that fills the pool in hours. How long will it
take to fill the pool using both hoses?
The length of time it would have taken for both garden hoses to fill the pool would be 6.67 hours.
How to find the time taken ?To solve this problem, we can use the formula:
1/rate = time
In the case of Joe's hose, the rate is 1/12 of the pool per hour. For his neighbor's hose, the rate is 1/15 of the pool per hour.
So, the combined rate is:
= (1/12) + (1/15)
= (5/60) + (4/60)
= 9/60 of the pool per hour
So, the time it will take to fill the pool using both hoses is:
= 1 / (9/60)
= 60/9 hours
= 6.7 hours
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The full question is:
Joe's garden hose fills the pool in 12 hours. His neighbor has thinner hose that fills the pool in 15 hours. How long will it take to fill the pool using both hoses?
(linear algebra)
(a) Let U be the subspace of C5 defined by U = {(z1, z2, z3, z4, z5) ? C5 ; 6z1 = z2 and z3+2z4+3z5 = 0} Find a basis of U.
(b) Extend the basis in part (a) to a basis of C5.
(c) Find a subspace W of C5 such that C5 = U direct sum W
a) One possible basis for U is {(1/6,1,0,0,0), (0,0,1,-2,3)}.
b) A basis of C5 is {(1/6,1,0,0,0), (0,0,1,-2,3), (1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}.
c) The basis of W is {(1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}.
Linear Algebra Identificationa) To find a basis of U, we can use the two given equations to solve for two of the variables in terms of the others. From the first equation, we have z1 = z2/6. Substituting this into the second equation, we get z3 + 2z4 + 3z5 = 0. Therefore, one possible basis for U is {(1/6,1,0,0,0), (0,0,1,-2,3)}.
b) To extend this basis to a basis of C5, we need to add three more vectors that are not in the subspace U. These vectors can be any linearly independent vectors that are not in U. One possible set of additional vectors is {(1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}. Therefore, a basis of C5 is {(1/6,1,0,0,0), (0,0,1,-2,3), (1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}.
c) To find a subspace W of C5 such that C5 = U direct sum W, we can take the complement of U in C5. The complement of U is the set of all vectors in C5 that are not in U. Since we have already found a basis for C5, we can simply take the subset of basis vectors that are not in U to form a basis for W. In this case, the basis of W is {(1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}.
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Julia sorted candies and found there were
10 each of these colors: green, red, and
pink and 20 blue in the package. Which
answer matches the percentage, decimal
and fraction of the blue candies in the
package?
F 20%, .2, 1/5
G 30%, .3, 3/10
H 40%, .4, 2/5
J 50%, .5, 1/2
The percentage is 40%, decimal point of blue candies is 0.4, and the fraction of blue candies is 2/5. The correct option is H. 40%, 0.4, 2/5.
How to calculate the percentage and decimal? Percentage is a unit of measurement used to express a proportion of a whole number in terms of hundredths. To calculate a percentage, divide the number of items in question by the total number of items and then multiply the result by 100. For instance, if you were to calculate the percentage of apples in a basket that contained 12 apples and 8 oranges, you would divide 12 by 20 (12 + 8) and then multiply the result by 100. The answer would be 60 percent.Decimals are numbers that are expressed in tenths, hundredths, or thousandths of a unit. To calculate a decimal, divide the number of items in question by the total number of items. For instance, if you were to calculate the decimal amount of apples in a basket that contained 12 apples and 8 oranges, you would divide 12 by 20 (12 + 8). The answer would be 0.6.Given,
Julia has 10 candies in green, 10 candies in red, 10 candies in pink and 20 candies in blue. There are 50 candies in Julia's hand.
We have to find percentage, decimal and fraction of blue candies = ?
Percentage = 20/50 x 100 = 40%
Decimal = 20/50 = 2/5 = 0.4
Fraction = 20/50 = 2/5.
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please help will mark BRAINLIEST
Answer:
90
Step-by-step explanation:
the bottom of the picture is cut off but it's 90°
becuase the picture shows you the 2 sides of the triangle r equal
the angles on the top are also equal
so that means the middle line has to be perpendicular to the base.
so Y is 90
The length of a new rectangular playing field is 4 yards longer than triple the width. If the perimeter of the rectangular playing field is 416 yards, what are its dimensions?
Answer: Let's call the width of the rectangular playing field "w" and the length "l". From the problem, we know that:
l = 3w + 4 (the length is 4 yards longer than triple the width)
We also know that the perimeter of a rectangle is the sum of the lengths of all four sides, so:
P = 2l + 2w
We are given that the perimeter is 416 yards, so we can substitute that into the perimeter equation:
416 = 2(3w + 4) + 2w
Simplifying and solving for w:
16 = 6w + 8
8 = 6w
w = 8/6 = 4/3
We can now substitute that back into the equation for the length:
l = 3w + 4
l = 3(4/3) + 4
l = 4 + 4
l = 8
So the dimensions of the rectangular playing field are 4/3 yards wide and 8 yards long.
Step-by-step explanation:
Find the exact circumference of each circle
The exact circumference for the circles are;
5. C = 94.26mi
6. C = 879.76in
7. C = 91.118in
8. C = 62.84m
9. C = 12.568m
10. C = 29.535cm
What is Circumference of a Circle?The circumference of a circle is its boundary or the length of the complete arc of a circle. The circumference of a circle is the perimeter of the circle. It is the total length of the boundary of the circle. The circumference of a circle is the product of the constant π and the diameter of the circle.
Circumference = πd
or
Circumference = 2πr
where π = 3.142 or 22/7
r = radius
d = diameter
5. C = 2πr
C = 2 x 3.142 x 15
C = 94.26mi
6. C = 2πr
C = 2 x 3.142 x 140
C = 879.76in
7. C = πd
C = 3.142 x 29
C = 91.118in
8. C = 2πr
C = 2 x 3.142 x 10
C = 62.84m
9. C = 2πr
C = 2 x 3.142 x 2.1
C = 12.568m
10. C = πd
C = 3.142 x 9.4
C = 29.535cm
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3) Six of the students in Ms. Collier's class play an
instrument. These six students make up 20% of
the class. How many students are in the class?
the base of a certain solid is the region r. cross sections through the solid taken perpendicular to the x-axis are squares. an integral that expresses the volume of the solid is:
The integral[tex]$\int_{-\infty}^{\infty} f(x)^2 dx$[/tex] expresses the volume of the solid, where f(x) is the side length of the square cross sections taken perpendicular to the x-axis in the region r.
The given integral [tex]$\int_{-\infty}^{\infty} f(x)^2 dx$[/tex] expresses the volume of the solid whose base is the region r. The integral is calculated by taking the cross sections of the solid perpendicular to the x-axis and consider them as squares. The side length of each square is denoted by f(x). The integral is calculated over the region r, which is the base of the solid. This integral is the sum of the areas of all the squares, which is equivalent to the volume of the solid. The integral is calculated by taking the lower limit of integration as[tex]$-\infty$[/tex] and the upper limit of integration as[tex]$\infty$[/tex]. Hence, the integral [tex]$\int_{-\infty}^{\infty} f(x)^2 dx$[/tex] expresses the volume of the solid with the base region r and square cross sections.
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