Which of the following double integrals would correctly solve this pro

Homework Statement

Which of the following double integrals would correctly solve this problem?

The Attempt at a Solution

I obtained two sets of boundary conditions.

Set 1:

$$x=-\sqrt{4-y^2}\quad (for\quad x<0)\quad to\quad x=\sqrt{4-y^2}\quad (for\quad x>0)\\y=-2\quad to\quad y=2$$

Set 2:

$$x=-2\quad to\quad x=2\\y=-\sqrt{4-x^2}\quad (for\quad y<0)\quad to\quad y=\sqrt{4-x^2}\quad (for\quad y>0)$$

This produces the following integrals:

$$\int_{-2}^{2}\int_{-\sqrt{4-y^2}}^{\sqrt{4-y^2}}(4-y)dxdy\\\int_{-2}^{2}\int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}(4-y)dydx$$

So why aren't a, b, and c all correct? The correct answer is c. Why are a and b incorrect?

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jedishrfu
Mentor
Looking at the diagram you want to choose the base correctly for symmetry of the sliced cylinder which is symmetrically split by the y-axis. So that means you would have to treat y as the independent variable and x as the dependent one so that gives you the x= sqrt(4-y^2).

That then implies the last choice C.