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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Use cylindrical shells to find the volume of the solid.

A right circular cone with height h and base radius r

To determine

To find:

The volume of the circular cone with the height h and the base radius r.

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 y- axis from a to b is

V= ab2πx f(x)dx

where,  0ab

2) Given:

The right circular cone with the height h and the base radius r

3) Calculation:

The right circular cone with the height h and the base radius r is as shown in the figure. It is obtained by rotating a triangle joining (0,0), (0, h) and (r,0) about y-axis

From the figure, write the equation of line passing through the two points (r, 0) and (0, h).

The slope of the line is =h-00-r The y-intercept is h

Therefore, the equation of the line is y=-hrx+h, and it rotates about y-axis.

Using the shell method, find the typical approximating shell with the radius x.

Therefore, the circumference is 2πx and the height is y=-hrx+h

So, the total volume is

V= ab2πx-hrx+hdx

V

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