Consider the region enclosed by the two right-opening parabolic arcs and

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

Problem

Consider the region enclosed by the two right-opening parabolic arcs

x=y^{2}

and

x=4-y^{2}

. This region is revolved about the

y

-axis. (a) Find all

y

-values where the curves

x=y^{2}

and

x=4-y^{2}

intersect. (b) For each

y

in the interval found in part (a), determine which curve is farther from the

y

-axis and hence identify the outer radius

R(y)

and inner radius

r(y)

for the generated solid. (c) Write a definite integral that represents the volume of the solid formed by the rotation, and evaluate the integral exactly. (d) Verify algebraically that the integrand in part (c) is always non-negative on the interval of integration.

Related theory

Area Between Curves
Area Applications
Volume Applications
Applications of Integration

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