Evaluate the triple integral of -2y dz dy dx where x runs from -3 to 3, y from 0 to the square root of 9 minus x squared, and z from 0 to 9 minus x squared minus y squared. Give an exact answer.

Step-by-step solution

1. 1

   Read the limits as a shape before integrating. The $x$ and $y$ limits describe $x^2+y^2\le 9$ with $y\ge 0$ — the upper half of the disk of radius $3$ — and $z$ runs from the plane $z=0$ up to the paraboloid $z=9-x^2-y^2$. Recognising the half-disk is what makes the cylindrical cross-check in the last step possible.

2. 2

   Integrate in $z$ first, because the integrand has no $z$. The inner integral is just the integrand times the height of the solid:

   $\int_{0}^{9-x^2-y^2}(-2y)\,dz=-2y\,(9-x^2-y^2)$

3. 3

   Integrate in $y$ and watch the two terms merge. Expanding, $-2y(9-x^2)+2y^3$, so with $R^2=9-x^2$:

   $\Bigl[-(9-x^2)y^2+\tfrac12 y^4\Bigr]_{0}^{\sqrt{9-x^2}}=-(9-x^2)^2+\tfrac12(9-x^2)^2=-\tfrac12(9-x^2)^2$

   The upper limit squares to exactly $9-x^2$, which is why the two powers of $R$ combine so cleanly.

4. 4

   Integrate in $x$ using the evenness of the integrand. Since $(9-x^2)^2=81-18x^2+x^4$ is even,

   $-\frac12\int_{-3}^{3}\!\!\bigl(81-18x^2+x^4\bigr)dx=-\int_{0}^{3}\!\!\bigl(81-18x^2+x^4\bigr)dx$
   $=-\Bigl[81x-6x^3+\tfrac{x^5}{5}\Bigr]_0^3=-\left(243-162+\frac{243}{5}\right)=-\frac{648}{5}$

5. 5

   Confirm the value in cylindrical coordinates. With $y=r\sin\theta$, $dV=r\,dz\,dr\,d\theta$, the same solid is $0\le r\le 3$, $0\le\theta\le\pi$, $0\le z\le 9-r^2$:

   $\int_0^{\pi}\!\!\int_0^{3}(-2r\sin\theta)(9-r^2)\,r\,dr\,d\theta=-2\left(\int_0^{\pi}\!\sin\theta\,d\theta\right)\left(\int_0^{3}\!r^2(9-r^2)\,dr\right)$

   The two factors are $2$ and $81-\tfrac{243}{5}=\tfrac{162}{5}$, giving $-2\cdot 2\cdot\tfrac{162}{5}=-\tfrac{648}{5}$ — the same exact value from a completely different route.

6. 6

   Interpret the sign. The whole solid sits in $y\ge 0$ and the integrand $-2y$ is $\le 0$ there, so a negative answer is expected; a positive result would immediately signal a sign slip.