A solid has base region lying on the -axis between and
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
A solid has base region T lying on the x-axis between x=−2 and x=3. The lower boundary of T is the x-axis. The upper boundary is piecewise defined by
y=x+3 for −2≤x≤0,
y=3 for 0≤x≤3.
At each x in [−2,3], a cross section perpendicular to the x-axis is constructed; the cross sections are rectangles whose height (in units) is given by h(x)=1+2e−x. Respond to the following prompts, showing complete justification.
(1) Part 1: (a) For −2≤x≤0, write an explicit expression for the base length b1(x) of a cross section.
(2) Part 2: (b) For 0≤x≤3, write an explicit expression for the base length b2(x) of a cross section.
(3) Part 3: (c) Formulate a definite integral for the volume V1 coming from the interval −2≤x≤0, and a separate definite integral for the volume V2 coming from the interval 0≤x≤3. Do not evaluate yet.
(4) Part 4: (d) Evaluate both integrals from part (c) exactly, and obtain the total volume of the solid. Express your answer in exact form, simplifying as far as possible.