A piece of glass artwork is created by rotating a planar region about the vertical line

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

Problem

x=4

. The region is bounded on the left by the

y

–axis, below by the

x

–axis, above by the curve

y=\dfrac{x^{2}}{2}

, and on the right by the vertical line

x=3

. Complete the tasks below to determine the volume of the finished piece. (a) Express the bounding parabola in the form

x

as a function of

y

, and determine the exact

y

–values that correspond to the lower and upper ends of the region when it is described using

y

as the independent variable. (b) Using

y

as the variable of integration, set up a definite integral that represents the volume of the solid obtained by revolving the region about

x=4

. (c) Evaluate the integral in part (b) to obtain the exact volume of the glass artwork.