   Chapter 5.3, Problem 27E

Chapter
Section
Textbook Problem

# Use the Midpoint Rule with n = 5 to estimate the volume obtained by rotating about the y-axis the region under the curve y = 1 + x 3 ,   0 ≤ x ≤ 1 .

To determine

To estimate:

The volume obtained by rotating about the y-axis the region under the given curve.

Explanation

1) Concept:

i. If x  is the radius of the typical shell, then the circumference =2πx and the height is y.

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

V= ab2πxf(x)dx

where,  0a<b

iii. To approximate the integral is the same as finding the Riemann sum for the midpoints. Use the formula and find the value of the integral.

2) Formula:

The Riemann sum for the midpoints:

Mn=i=1ng(xi-)x=abg(x)dx

where, xi-=12xi-1+xi and  x=b-an, n is the number of the subintervals. And xi=a+ix

3) Given:

y=1+x3, 0x1

4) Calculation:

As the region is bounded by

y=1+x3, 0x1

Draw the region using the given curves.

The graph shows the region and the height of cylindrical shell formed at x by the rotation about the line  y-axis

Therefore, the circumference is 2πx and the height is 1+x3

It is given that 0x1

So, a=0 and b=1

So, the volume of given solid is

V=012πxfxdx

=012πx1+x3dx

Assume that

V=01gxdx

where gx=2πx1+x3

For the midpoint rule,

Find  x &xi-

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