Let be the region in the first quadrant bounded by the graphs of and between and

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

Problem

Let

S

be the region in the first quadrant bounded by the graphs of

y=\sin x

and

y=\cos x

between

x=0

and

x=\dfrac{\pi}{2}

. The region

S

is revolved about the horizontal line

y=-1

to generate a solid. (a) Determine the

x

–coordinate where

y=\sin x

and

y=\cos x

intersect on

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

. Justify your answer. (b) Express the volume of the solid as the sum of two definite integrals, each corresponding to an interval on which one curve remains consistently above the other. Clearly indicate the limits of integration in each integral. (c) Evaluate the integrals from part (b) to obtain the exact volume of the solid. (d) Interpret the physical meaning of the quantity

V'(k)

, where

V(k)

denotes the volume of the solid obtained when the region

S

is revolved about

y=-k

for

k>0

. Include units in your explanation.

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