Evaluate $\iint_{D}(x^2+y^2)\,dA$ where $D=\{x^2+y^2\le 4,\ y\ge x\}$.
1
300 sec
Q.
Reverse the order: $\int_{x=0}^{2}\int_{y=x}^{2x} f(x,y)\,dy\,dx$ equals:
2
300 sec
Q.
For $u=x-y,\ v=xy$ restricted to $x\ge y>0$, compute $\left|\dfrac{\partial(x,y)}{\partial(u,v)}\right|$ in terms of $(u,v)$.
3
300 sec
Q.
For the surface $x^2+2y^2+3z^2=14$ at $(2,1,1)$, the **normal line** is:
4
300 sec
Q.
Lamina in first quadrant bounded by $x^2+y^2\le 4$, $y\le x$; density $\rho=x$. The center of mass $(\bar{x},\bar{y})$ is:
5
300 sec
Q.
For the solid sector $0\le r\le 1$, $0\le \theta\le \tfrac{\pi}{2}$, $0\le z\le 2$ with density $\rho=1+r$, compute $I_z=\iiint r^2\rho\,dV$.
6
300 sec
Q.
Let $\mathbf{F}=\langle y^2+x,\ x^2-y\rangle$ and $C$ the boundary of $[0,1]\times[0,2]$ (CCW). Compute $\oint_C \mathbf{F}\cdot d\mathbf{r}$.
7
300 sec
Q.
Evaluate the line integral $\displaystyle \oint_C (2x - y)\,dx + (x + 3y)\,dy$ where $C$ is the boundary of the region bounded by the lines $y = 0$, $x = 2$, and $y = x$, taken in the counterclockwise direction.
8
300 sec
Q.
Evaluate $\iiint_R z\,dV$ where $R$ is inside the sphere $x^2+y^2+z^2\le 9$ and above the cone $z=\sqrt{x^2+y^2}$.
9
300 sec
Q.
Choose correct limits for $\iiint_R 1\,dV$ where $R=\{(x,y,z): x^2+y^2\le 4x,\ 0\le z\le x\}$.
