Consider the region under the line on the interval and above the x-axis

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

Problem

Consider the region

R_0

under the line

y = x

on the interval

0 \le x \le 3

and above the x-axis. The region is revolved about the horizontal line

y = 2

, producing a solid. (1) Part 1: (a) Determine the

x

-value where the line

y = x

intersects the axis of rotation

y = 2

. (2) Part 2: (b) Explain why the solid must be decomposed into two parts, one for

0 \le x \le 2

and one for

2 \le x \le 3

, when writing an expression for its volume. (3) Part 3: (c) Write the two definite integrals with respect to

x

that, together, give the volume of the solid. (4) Part 4: (d) Evaluate the integrals and state the exact volume.