   Chapter 10, Problem 26RE

Chapter
Section
Textbook Problem

# Using an Ellipse Consider the ellipse x 2 25 + y 2 9 = 1 .(a) Find the area of the region bounded by the ellipse.(b) Fund the volume of the solid generated by revolving the region about its major axis.

To determine

To calculate:

(a) The area of the region bounded by the ellipse whose equation is x225+y29=1.

(b) The volume of the solid generated by revolving the region around the major axis of the ellipse x225+y29=1.

Explanation

Given:

The equation of ellipse is x225+y29=1.

Calculation:

(a)

Consider the equation.

x225+y29=1

Compare the equation x225+y29=1 with the standard equation of an ellipse

(xh)2a2+(yk)2b2=1

So, a2=25, b2=9

Then,

a=5.

b=3

As a>b so the ellipse is of the type horizontal major axis and the graph of ellipse is

The area of the region bounded by the ellipse is given by

A=πab

Then,

A=π×5×3=15π

(b)

The equation of the ellipse is: x225+y29=1

Here, the major axis is along the x –- axis.

Rearrange the equations as:

y29=(1x225)y2=9(1x225)=925(25x2)

The required volume is double the volume of the solid obtained by revolving the area in the first quadrant

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