Let be the region in the first quadrant bounded above by the graph of and below by…
MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
Let R be the region in the first quadrant bounded above by the graph of y=4 and below by the graph of y=x2, with the additional boundary x=0. The region R is revolved about the horizontal line y=6 to generate a solid S.
(1) Part 1: (a) Describe, without performing any calculations, which variable of integration (x or y) leads to a single integral that is simpler to evaluate for the volume of S, and justify your choice by referring to the geometry of the solid.
(2) Part 2: (b) Using your chosen variable of integration, write, but do not evaluate, a definite integral expression for the volume of S.
(3) Part 3: (c) Evaluate the integral from part (b) to obtain the exact volume of the solid S.