Find the volume of the solid analytically.

Question

Calc AB, Unit 8.3

The solid lies between planes perpendicular to the x-axis at x = -1 and x =1. The cross sections perpendicular to the x-axis are circular disks whose diameters run from the parabola y = x^2 to the parabola y = 2 - x^2.

Answer 1

So Lissie A.,

Draw the two boundary functions (as an x-y plane diagram), then a diagram of a single disc, at an angled view. It has the shape either of a solid disc (but only at x=0) or an annulus (a circle minus an inner circle). So write the expression for the area of that annulus (= area total circle minus area inner missing circle) That is a polynomial in x, isn't it. Then integrate that expression from x=[-1,+1]. You know how to integrate functions like constants, x's, and x^2's, right (not saying if you'll have all of those, just reminding you that you do know how!)? Note that the indefinite_integral_added_constant drops out when you integrate over a specified domain.

-- Cheers, --Mr. d.