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

Evaluate the line integral of the vector field $F(x,y)=(x+7y,x^2)$ along the curve $C$, where $C$ is the line segment from $(0,0)$ to $(1,2)$. Which of the following is the value of the integral ${\int }_C^{}(x+7y)dx+x^{2}dy$?

Which of the following statements is true about the function $g$ defined on the closed interval $[-4,8]$?

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?

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

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 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.