   Chapter 5.3, Problem 34E

Chapter
Section
Textbook Problem

# Use a graph to estimate the x-coordinates of the points of intersection of the given curves. Then use this information and your calculator to estimate the volume of the solid obtained by rotating about the y-axis the region enclosed by these curves. y = 3 sin x ,     y = x 2 − 4 x + 5

To determine

To find:

i. x-coordinates of the points of intersection

ii. The volume of the solid

Explanation

1) Concept:

Let  S be a solid.

By the shell method, the volume of the solid by rotating the region bounded by the curve y=f(x) about y-axis from a to b is

V= ab2πxf(x)dx

where 0ab

2) Given:

y=x2-4x+5,  y=3sinx

3) Calculations:

Given,

y=x2-4x+5,  y=3sinx

First, draw the graph of the given curves.

i. From the graph of the given curves,

The  x coordinate of the points of intersection is x=0.866 and x2.648

ii. Also,x2-4x+5<3sinx on the interval (0.866, 2.648)

Therefore, the height of typical shell is   3sinx-(x2-4x+5)

So, the volume of the solid rotating about y-axis is

V= ab2πxf(x)dx

where,  0ab

that is,

V= 0

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