   Chapter 5.2, Problem 34E

Chapter
Section
Textbook Problem

# Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. y = x 2 ,   x 2 + y 2 = 1 ,   y ≥ 0 (a) About the x-axis(b) About the y-axis

To determine

Part (a):

i. To set up:

An integral for the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis

ii. To evaluate:

The integral by using the calculator up to five decimal places

Explanation

1) Concept:

i. If the cross section is a washer with inner radius rin and outer radius rout, then area of the washer is obtained by subtracting the area of the inner disk from the area of the outer disk,

ii. The volume of the solid revolution about x-axis is

V= abA(x)dx

2) Given:

y=x2, x2+y2=1, y0, about the x-axis

3) Calculation:

Region bounded by the given curves is shown below:

The region rotates about the  x-axis, so the cross-section is perpendicular to the x-axis and it forms a washer.

To find the outer and inner radius, solve the given equation for y because limits of integration are in terms of x

x2+y2=1

Subtract x2 from both sides,

y2=1-x2

By taking square root of both sides,

y=1-x2  …………….………….. since y0

The outer radius is the distance from the  upper part of the circle to the axis of rotation x axis

The inner radius is the distance from the curve to the axis of rotation x axis

To find limits of integration equate

To determine

Part (b):

i. To set up:

An integral for the volume of the solid obtained by rotating the region bounded by the given curves about the y-axis

ii. To evaluate:

The integral by using the calculator up to five decimal places

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