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

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

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

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 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?

Evaluate the line integral of the vector field $F(x,y)=(y,x)$ along the curve $C$, which is the line segment from $(0,0)$ to $(1,1)$.

Let C be the curve parameterized by $x=t^2$, $y=t^3$ for $0\le t\le 1$. Find the value of the line integral ${\int }_C^{}ydx$.

Find the exact length of the curve $y=\frac{1}{4}{x}^2-\frac{1}{2}\ln \left(x\right)$ for $1\le x\le 2$.