Region is bounded above by the graph of and below by the -axis for
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Region R is bounded above by the graph of y=4−x2 and below by the x-axis for −2≤x≤2. A solid is constructed so that for each x in [−2,2], the cross section taken perpendicular to the x-axis is an isosceles right triangle whose leg lies entirely in R. (The right angle is at one end of the leg.) Assume all triangles are congruent in orientation.
Answer the following questions about the solid.
(a) Express the length of the leg of a typical triangular cross section at a fixed value of x in terms of x. Clearly justify your expression in the context of region R.
(b) Using your result from part (a), explain how the area of the triangular cross section depends on x. Your explanation should reference appropriate geometric properties of an isosceles right triangle.
(c) Describe, without carrying out the computation, how the definite integral representing the volume of the solid is constructed from the information in parts (a) and (b). Be sure to specify the limits of integration and the integrand in words or symbols and to justify each element of your integral setup.