Find the exact length of the curve $y=\frac{2}{3}(1+x^2){(}^{\frac{3}{2}}$ for $0\le x\le 4$.

Evaluate the double integral ${\int }_0^2{\int }_0^1(2x+1+xy) dy dx$.

Calculate the double integral of $f(x,y)=x+y$ over the region $R$, where $R$ is the rectangle defined by $0\le x\le 2$ and $1\le y\le 3$.

Determine the area of the region bounded by $y=x+\sin x$, the x-axis, $x=0$, and $x=\pi$.

Evaluate the definite integral: ${\int }_{-2}^3(x^2-3) dx$.

Calculate the value of the double integral ${\int }_0^4{\int }_0^1(4x+1+xy) dy dx$.

Find the area under the curve $y=x^2$ from $x=1$ to $x=3$.

Given the graph of $f\left(x\right)$ below, evaluate the definite integral ${\int }_0^{4}f\left(x\right)dx$.

Given the parametric equations $x={\sin }^2\left(t\right)$, $y=8\cos \left(t\right)$ for $0\le t\le \pi$, what is the area enclosed by the curve and the y-axis?