Let be the region in the first quadrant bounded by ,

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Let

R

be the region in the first quadrant bounded by

y = \sin x

y = \cos x

, and the

y

-axis (

x = 0

). Region

R

is revolved about the horizontal line

y = -1

to form a solid. (a) Find the

x

–coordinate in the interval

0 \le x \le \frac{\pi}{2}

at which the graphs of

y = \sin x

and

y = \cos x

intersect. (b) On each sub-interval of

[0, \frac{\pi}{2}]

determined in part (a), write expressions for the outer radius and the inner radius of the washers in terms of

x

. (c) Write a piecewise definite integral expression for the exact volume of the solid. (d) Evaluate the integral(s) from part (c) to obtain a closed-form expression for the volume. (e) Using your result from part (d), approximate the volume to three decimal places.

Related theory

Area Between Curves
Area Applications
Volume Applications
Applications of Integration

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