Evaluate the definite integral: ${\int }_0^{\frac{\pi }{4}}\sin \left(t\right){\cos }^2\left(t\right) dt$.

Find the area of the region enclosed by one loop of the curve $r=\sin \left(8\theta \right)$.

Evaluate the definite integral: ${\int }_0^{\frac{\pi }{6}}\sin \left(t\right){\cos }^2\left(t\right) dt$.

Evaluate the definite integral: ${\int }_1^4(t^3-t^2-4) dt$.

Evaluate the double integral ${\int }_D^{}(2x+y)da$, where $D=\left\{\right(x,y)\mid 1\le y\le 4,y-3\le x\le 3\}$.

Evaluate the line integral ${\int }_C^{}{y}^{3}ds$, where $C$ is the curve given by $x=t^3$, $y=t$, for $0\le t\le 4$.

Evaluate the definite integral: ${\int }_1^5{w}^2\ln \left(w\right) dw$

Find the exact length of the curve given by $x=5\cos \left(t\right)-\cos \left(5t\right)$, $y=5\sin \left(t\right)-\sin \left(5t\right)$, for $0\le t\le 2\pi$.

Evaluate the iterated integral: ${\int }_0^4{\int }_0^{y^2}{x}^{2}ydxdy$