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 ${\int }_C^{}(x+7y)dx+x^{2}dy$?

Find the third-degree Taylor polynomial, $t_3\left(x\right)$, for the function $f\left(x\right)=\frac{1}{x}$ centered at $a=2$.

Given the polar curve $r=\cos \left(2\theta \right)$, what is the exact length of one petal of the curve?

Given the Laplace transform $F\left(s\right)=\frac{1}{s^2+1}$, find the corresponding function $f\left(t\right)$.

Evaluate the line integral of the vector field $F(x,y)=(x+6y)dx+x^{2}dy$ along the curve $C$, where $C$ is the line segment from $(0,0)$ to $(2,1)$. Which of the following is the value of the integral?

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?

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