Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Use the shell method to find the volume of the solid generated by revolving the shaded region about the
x-axis.
Set up the integral that gives the volume of the solid.
The volume of the solid generated by revolving the shaded region about the
x-axis is
◻ cubic units. (Type an exact answer, using
pi as needed.)

Use the shell method to find the volume of the solid generated by revolving the shaded region about the $x$ -axis. $\newline$ Set up the integral that gives the volume of the solid. $\newline$ The volume of the solid generated by revolving the shaded region about the $x$ -axis is $\square$ cubic units. (Type an exact answer, using $\pi$ as needed.)

View step-by-step help

View step-by-step help

Evaluate two-variable equations: word problems

Full solution

Q. Use the shell method to find the volume of the solid generated by revolving the shaded region about the

x

-axis.

\newline

Set up the integral that gives the volume of the solid.

\newline

The volume of the solid generated by revolving the shaded region about the

x

-axis is

\square

cubic units. (Type an exact answer, using

\pi

as needed.)

Identify function & limits: Identify the function and limits for the shaded region. Assume the function is $y = x^2$ from $x = 0$ to $x = 2$ .
Set up integral: Set up the integral using the shell method. The radius of each cylindrical shell is $x$ , and the height is the function value $y = x^2$ . The volume of each shell is approximately $2\pi xy \, dx$ .
Write integral for volume: Write the integral for the volume: $V = \int_{0}^{2} 2\pi x(x^2) \, dx$ . Simplify the integrand to $2\pi x^3$ .
Calculate integral: Calculate the integral: $V = 2\pi \int_{0}^{2} x^3 \, dx$ .
Solve integral: Solve the integral: $\int x^3 \, dx = \frac{x^4}{4}$ . Evaluate from $0$ to $2$ , which gives $\left(\frac{2^4}{4}\right) - \left(\frac{0^4}{4}\right) = \frac{16}{4} - 0 = 4$ .
Multiply by $2\pi$ : Multiply by $2\pi$ : $V = 2\pi \times 4 = 8\pi$ cubic units.

More problems from Evaluate two-variable equations: word problems

Question

This equation shows how the distance traveled by a car is related to the time spent driving.

d = 50t + 10

t

represents the time in hours, and the variable

d

represents the distance in miles. How far does the car travel in

2

Posted 4 months ago

Question

The equation shows how the cost of buying pencils is related to the number of pencils purchased.

\newline

c= 0.5p+2

\newline

p

represents the number of pencils, and the variable

c

represents the total cost. How much does it cost to buy

10

Posted 4 months ago

Question

This equation shows how the total score in a game is related to the number of points scored per level.

s = 100l + 250

l

represents the levels completed, and the variable

s

represents the total score. What is the total score after completing

3

Posted 2 months ago

Question

The equation shows how the amount of paint needed is related to the area to be painted.

p = 2a + 5

a

represents the area in square meters, and the variable

p

represents the amount of paint in liters. How much paint is needed for an area of

8

Posted 2 months ago

Question

This equation shows how the total bill for a meal is related to the number of people sharing the meal.

\newline

b = 15n + 35

\newline

n

represents the number of people, and the variable

b

represents the total bill. What is the total bill for a meal shared by

4

Posted 4 months ago

Question

Poppy the dog weighs

p

pounds, and Yumi the cat weighs

y

pounds. If Poppy weighs

36

more pounds than Yumi does, which of the following equations correctly describes the relationship between their weights?

\newline

1

\newline

p+y=36

\newline

p+36=y

\newline

p-y=36

\newline

y-p=36

Posted 1 month ago

Question

Calvin and Melvin are two of Santa's elves. The number of toys they build is given by

11 c+8 m

c

is the number of candy canes Calvin eats for breakfast and

m

is the number of candy canes Melvin eats for breakfast. How many toys do they build when Calvin eats

5

candy canes and Melvin eats

3

candy canes for breakfast?

Posted 1 month ago

Question

2 w+4

\newline

Carly is cutting a piece of paper stock based on the width of a photo. The expression above describes the length of the paper stock, in inches, for a photo that is

w

inches wide. If the photo is

8

5

inches wide, what is the length of the paper stock in inches?

Posted 1 month ago

Question

Calvin and Melvin are two of Santa's elves. The number of toys they build is given by

11 c+8 m

c

is the number of candy canes Calvin eats for breakfast and

m

is the number of candy canes Melvin eats for breakfast.

\newline

How many toys do they build when Calvin eats

5

candy canes and Melvin eats

3

candy canes for breakfast?

\newline

Posted 2 months ago

Question

The approval rating of a certain politician is changing at a rate of

r(t)

percent of the population per month (where

t

is the time in months).

\newline

\int_{0}^{3} r(t) d t

\newline

1

\newline

(A) The rate of change of the approval rating of the politician when

t=3

\newline

(B) The average rate of change of the approval rating during the first three months

\newline

(C) The change in the percent of the population who approves of the politician during the first three months

\newline

(D) The percent of the population who approves of the politician when

t=3

Posted 1 month ago