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: $${\int }_0^2{\int }_{2-x^2}^{4-x^2}{x}^2+y^2 dy dx$$