So I have had Calc III many moons ago but I cannot seem to solve this problem for my son who is taking it now. Worked it four ways and got four different answers. Hoping someone here can set me right.
Problem: Solve $\iint f(x,y)\,dS,$ where
$$f(x,y) = x + y$$ and $$S(u,v) = 2\cos(u)\vec i + 2\sin(u)\vec j + v \vec k$$ $$0\le u \le \pi/2;\;0 \le v \le 9. $$
My solution is: $$\frac{\partial S}{\partial u} = -2\sin(u) \vec i + 2\cos(u) \vec j$$ $$\frac{\partial S}{\partial v} = \vec k$$ $$\frac{\partial S}{\partial u} \times \frac{\partial S}{\partial v} = 2\cos(u) \vec i - 2 \sin(u) \vec j $$ $$\left\lvert\frac{\partial S}{\partial u} \times \frac{\partial S}{\partial v} \right\rvert = \sqrt{4\cos^2(u) + 4 \sin^2(u)}=2$$ $$\iint(x+y)\,dS =\iint(x+y)2\,dA$$ $$\int^{\pi/2}_0\int^2_0(r\cos\theta +r\sin\theta) 2r\,dr\,d\theta =$$ $$\int^{\pi/2}_0\int^2_02r^2(\cos\theta +\sin\theta) \,dr\,d\theta =$$ $$\int^{\pi/2}_0(16/3)(\cos\theta +\sin\theta) \,d\theta =$$ $$32/3.$$
His online program doesn’t like this answer. I got the same answer by staying in rectangular coordinates. Feel like I’m setting something up wrong.
Thanks in advance for any help.