The base of a solid is the interior of the ellipse

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

Problem

\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}\le 1

. At each value of

y

in the base, the cross section perpendicular to the

y

-axis is a rectangle whose height is three times its base (width). (a) Write an expression for half of the horizontal width of the base as a function of

y

. (That is, find

x_{\text{right}}(y)

for

y

in the interior of the ellipse.) (b) Using your result from part (a), obtain a formula for the area

A(y)

of a typical rectangular cross section in terms of

y

. Simplify completely. (c) Set up and evaluate a definite integral that gives the exact volume of the solid.

Related theory

Ellipse
Conic Sections
Volume Applications
Applications of Integration

Browse the problem library