0. Functions

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

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

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 definite integral: ${\int }_0^1\frac{2}{4}{t}^3-\frac{3}{4}{t}^2+\frac{2}{5}tdt$ from $t=0$ to $t=1$.

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

Evaluate the integral: ${\int }_{x=1}^{x=2}2x(x^2+4x+20) dx$.

Evaluate the iterated integral by converting to polar coordinates: $${\int }_0^2{\int }_{2-x^2}^{4-x^2}{x}^2+y^2 dy dx$$

Find the exact length of the curve given by $x=\frac{y^4}{8}+\frac{1}{4y^2}$, for $1\le y\le 3$.

Is the constant function $y\left(t\right)=-4$ a solution to the differential equation $y^{\prime }=t^2(y+4)$?

Which of the following is a power series representation for the function $f\left(x\right)=x^6\arctan \left(x^3\right)$ centered at $x=0$?

Which of the following is a power series representation for the function $f\left(x\right)=x^4\arctan \left(x^3\right)$ centered at $x=0$?

Which of the following is a power series representation for the function $f\left(x\right)=x^2\arctan \left(x^3\right)$ centered at $x=0$?

Find the exact length of the curve defined by $x=\frac{1}{3}y(y-3)$ for $9\le y\le 16$.

Given the boundary-value problem $y^{″}-2y^{\prime }+2y=2x-2$, with $y\left(0\right)=0$ and $y\left(\pi \right)=\pi$, which of the following is the correct solution for $y\left(x\right)$?

For the function $f$ whose graph is given, which of the following best describes the domain of $f$?