   Chapter 15, Problem 39RE

Chapter
Section
Textbook Problem

Find the volume of the given solid.39. One of the wedges cut from the cylinder x2 + 9y2 = a2 by the planes z = 0 and z = mx

To determine

To find: The volume of the given solid.

Explanation

Given:

The region E is the wedge cut from the cylinder x2+9y2=a2 by the planes z=0,z=mx .

Formula used:

The volume of the given solid is given by, V=EdV .

Calculation:

Solve the given equation as given below.

z=mxmx=0x=0(0)2+9y2=a2

Simplify this further as given below.

9y2=a2y2=(a3)2y=±(a3)2y=±(a3)

From the above equation and from the given conditions, it is observed that x varies from 0 to a29y2 and y varies from a3 to a3 . Then, the volume of the given solid is,

V=EdV=a3a30a29y2mxdxdy

Integrate with respect to x and apply the limit.

V=a3a3[mx22]0a29y2dy=a3a3[m(a29y2)22m(0)22]dy=

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