Evaluate the line integral of the vector field $F(x,y)=(2x,3y)$ along the path $C$ given by $r\left(t\right)=(t,t^2)$ for $0\le t\le 1$.

Evaluate the integral: ${\int }_0^{\frac{\pi }{2}}\frac{\cos \left(t\right)}{1+{\sin }^t^2}dt$

Evaluate the triple integral over the region E, where E lies above the cone $z=\frac{r}{3}$ and below the sphere $z^2+r^2=1$, of the function $y^2{z}^2$ $dV$. Which of the following is the correct value of the integral?

Find the area of the region enclosed by one loop of the curve $r=\sin \left(4\theta \right)$.

Which of the following is the correct definition of a function in calculus?

Evaluate the vector integral from $t=0$ to $t=4$ of the vector function $sec\left(t\right)tan\left(t\right)\mathbf{i}+tcos\left(2t\right)\mathbf{j}+{sin}^2\left(2t\right)cos\left(2t\right)\mathbf{k}$ with respect to $t$.

Given the power series $\sum \underset{n=1}{}\stackrel{\infty }{}\frac{{(}^{-1}n}{x^n}/9^n$, find the radius of convergence, $r$, of the series.

Evaluate the triple integral of $1$ over the part of the ball defined by $x^2+y^2+z^2\le 8$ that lies in the first octant.

Evaluate the triple integral ${\int }_{}^{}{\int }_{}^{}{\int }_{}^{}z{d}_V$, where E is the region bounded by the parabolic cylinder $y=x^2$, the planes $z=0$, $z=4$, and $y=4$.