I need to calculate the area of the solid $$S=\{(x,y,z): x,y\geq 0, x^2+y^2\leq 1, 0\leq z \leq xy \}.$$ I'm not aksing you to do it for me, I'm just asking to check whether I did it right. (I would be very very glad, if you could answer quickly, since I need this for a test.)
I did this by calculating the area for each of the $5$ sides of $S$ separately. I'm especially unsure if calculated one of the the lateral sides correctly. I'm only illustrating here how I calculated the area of the side $$S_1=\{(x,y,z): x,y\geq 0, x^2+y^2=1, 0\leq z \leq xy \}.$$Please tell me, if the calculation and the steps I took are correct:
We have via a cylindrical coordinate transformation that $$S_1=\{(1,\phi,v): \phi \in [0,\frac{\pi}{2}], 0 \leq v \leq \cos\phi\sin\phi\},$$so $S_1=F(K)$ for $F:K\rightarrow \mathbb{R^3}$, $F(\phi,v)=(1,\phi,v)$ and $$K=\{(\phi,v): \phi \in [0,\frac{\pi}{2}], 0 \leq v \leq \cos\phi\sin\phi\}.$$Now we can use this parametrization to calculate the area with the iterated integral:$$\int_0^{\frac{\pi}{2}}\int_0^{\cos\phi\sin\phi} \sqrt{\det(\operatorname{D}\phi)^{\operatorname{tr}}\operatorname{D}\phi}\,dv\,d\phi=\frac{1}{2}.$$
(If someone has some spare time and wants to also verify the volume of $S$: I calculated that the volume of $S$ is $\frac{1}{8}$. Is that the right value ?)