Evaluate $\iint_{R} (x+2y)\,dA$, where $R$ is the triangle with vertices $(0,0),(2,0),(0,4)$.

Reverse the order of integration: $\int_{y=0}^{1}\int_{x=y}^{1} f(x,y)\,dx\,dy$ equals:

For the transformation $u=x^2-y^2,\ v=2xy$, the absolute Jacobian $\left|\dfrac{\partial(x,y)}{\partial(u,v)}\right|$ equals:

Evaluate $\oint_C \mathbf{F}\cdot d\mathbf{r}$ for $\mathbf{F}=\langle -y^3,\ x^3\rangle$ where $C$ is the unit circle oriented counterclockwise.

For $\mathbf{F}=\langle x^2y,\ yz,\ zx\rangle$, compute $\nabla\cdot\mathbf{F}$ at $(1,2,3)$.

Evaluate $\iiint_{1\le \rho\le 2} \rho^2\, dV$ in spherical coordinates.

Let $\mathbf{F}=\langle 2xy+e^x,\ x^2+2y\rangle$. Compute the work $\int_C \mathbf{F}\cdot d\mathbf{r}$ from $(0,0)$ to $(1,1)$ along any path.

Find the area of the region enclosed by $y=x^2$ and $y=2x$.

Let $C$ be the ellipse $\dfrac{x^2}{4}+y^2=1$ oriented CCW. Evaluate $\dfrac{1}{2}\oint_C (x\,dy - y\,dx)$.

Choose the correct limits for $\iint_R 1\,dA$ where $R$ is bounded by $y=x^2$ and $y=1+x$.