Let be the solid whose base is the region in the first quadrant bounded by the -axis

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

Problem

Let

S

be the solid whose base is the region in the first quadrant bounded by the

x

-axis, the line

x=3

, and the parabola

y=x^2

. For each

x

between

0

and

3

, the cross section of

S

perpendicular to the

x

-axis is a rectangle. The width of the rectangle equals the

y

-value of the base at that

x

, and the height of the rectangle is

h(x)=x

. Units are centimeters. (a) Set up, but do not evaluate, a definite integral that represents the volume

V

of the solid. (b) Evaluate the integral from part (a) to find the exact volume of

S

. (c) Approximate the volume to the nearest cubic centimeter.