What is the volume of the surface intercepted by x^2+y^2=1 , z=2-|x| , z=0 ?

Question

triple integrals

Answer 1

Let's first think about what this solid looks like, and hopefully we can use some symmetry to make things easier. The base is a circle on the xy-plane. The top is bounded by z = 2-|x|, so it will have an edge at z=2 along the y-axis, and slope downward on either side as |x| increases, like a chisel edge. You can either split this into 2 parts since |x| is a piecewise function, or integrate on one side of the chisel and double it.

 

The easiest half to take is the half on the +x side, so you don't have to worry about sign errors nearly as much.

 

The limits of the integrals will be:

y from -1 to 1

x from 0 to √(1-y²)  [The +x side of the circle]

z from 0 to 2-x

 

Since z depends on x and x depends on y, you'll need to integrate in that order.

 

Also, since the base is a circle, you may want/need to cut it into a quarter, just the first xy-quadrant:

 

4 ∫₀¹∫₀^{√(1-y^2)}∫₀^2-x dz dx dy

 

Hopefully you can do it from here. Please comment if you have further questions!