Let be the region in the first quadrant bounded by and

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

Problem

Let

T

be the region in the first quadrant bounded by

y=x^{3}

and

y=3x^{2}

. The region is revolved about the horizontal line

y=30

. (a) Find all points of intersection of

y=x^{3}

and

y=3x^{2}

for

x\ge 0

. (b) On each subinterval determined by the intersection points, identify which curve is farther from the axis of revolution

y=30

. Justify your answer analytically. (c) Write, but do not evaluate, an expression (which may require more than one definite integral) for the volume obtained by revolving

T

about

y=30

. (d) Evaluate the expression from part (c) to obtain the exact volume of the solid.

Related theory

Area Between Curves
Area Applications
Volume Applications
Applications of Integration

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