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Chapter 6: Q. 3 (page 509)

Consider the rectangle bounded by y = 3 and y = 0 on the x-interval [2,2.5].

(a) What is the volume of the disk obtained by rotating this rectangle around the x-axis?

(b) What is the volume of the washer obtained by rotating this rectangle around the line y=5?

Short Answer

Expert verified

(a) The volume of the disk obtained by rotating this rectangle around the x-axis is 2.25π.

(b) The volume of the washer obtained by rotating this rectangle around the line y=5 is 5.25π.

Step by step solution

01

Part (a) Step 1. Given Information.

The rectangle bounded by y = 3 and y = 0 on[2,2.5]

02

Part (a) Step 2. Volume of the disk.

The volume of the disk formed by rotating around the x-axis is given by,

V=∫ab[r(x)]2dx where r(x)is the radius of the disk.

From the rectangle, r(x)=3

03

Part (a) Step 3. Volume of the disk.

The volume of the disk is determined by,

V=π∫22.532dx=9π∫22.25dx=9π[x]22.25=9π[2.25-2]=2.25π

04

Part (b) Step 1. Draw the region.

Draw the region along with the rotation on the coordinate plane.

05

Part (b) Step 2. Volume by Washer method.

The volume of washer formed is given by the formula,

V=∫ab[[R(x))2-(r(x))2]dx

R(x)=5;r(x)=2

06

Part (b) Step 3. Find the volume.

The volume is,

V=π∫22.2552-22dx=21π∫22.25dx=21π[x]22.25=5.25π

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