# Volume of Solid Revolving Region Bounded by x=y^2, x=4 About x=5

• Blonde1551
In summary, the conversation is about finding the volume of a solid generated by revolving a region bounded by two curves around a given line. The washer method is used, with r(y) representing the inner radius and R(y) representing the outer radius. The correct approach is to set r(y) = 0 and R(y) = y^2, and the resulting integral should be ∏∫(4-y^4) from 0 to 2, giving an answer of 16.96.
Blonde1551

## Homework Statement

Find volume of the solid generated by revolving the region bounded by x = y^2, x=4
about the line x = 5

## Homework Equations

the washer method from c to d ∏∫ R(y)2 - r(y)2

## The Attempt at a Solution

I set r(y) = 1 and R(y)= y^2

and got the integral from 0 to 2 of ∏∫(y^2)^2-(1)^2

I got an answer of 16.96 but i know this is wrong because the back of the book gives a different answer. Please tell me where I went wrong.

Last edited:
Blonde1551 said:

## Homework Statement

Find volume of the solid generated by revolving the region bounded by x = y^2, x=4
about the line x = 5

## Homework Equations

the washer method from c to d ∏∫ R(y)2 - r(y)2

## The Attempt at a Solution

I set r(y) = 1 and R(y)= y^2

and got the integral from 0 to 2 of ∏∫(y^2)^2-(1)^2

I got an answer of 16.96 but i know this is wrong because the back of the book gives a different answer. Please tell me where I went wrong.
Think about what the washer method involves -- why it's called the "washer" method.

What do r(y) and R(y) represent?

## 1. What is the formula for finding the volume of a solid revolving region bounded by x=y^2, x=4 about x=5?

The formula for finding the volume of a solid revolving region bounded by x=y^2, x=4 about x=5 is V = π∫(4-y^2)^2 dy, where the limits of integration are from y=0 to y=2.

## 2. What does the variable y represent in this formula?

The variable y represents the distance from the axis of revolution (x=5) to the curve x=y^2.

## 3. How do you determine the limits of integration for this problem?

The limits of integration are determined by finding the intersection points between the two equations x=y^2 and x=4. These intersection points are y=0 and y=2, which become the lower and upper limits of integration, respectively.

## 4. Can this formula be used for any solid revolving region?

No, this formula is specifically for a solid revolving region bounded by x=y^2, x=4 about x=5. Different shapes and boundaries will require different formulas for finding the volume.

## 5. How does changing the axis of revolution affect the volume of the solid?

Changing the axis of revolution can greatly affect the volume of the solid. In this problem, changing the axis from x=5 to any other value would result in a different volume. In general, the volume of a solid revolving region is proportional to the distance from the axis of revolution.

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