Evaluate the line integral of the function $f(x,y)=xy$ along the curve $C$, where $C$ is given by the parametric equations $x=t^2$, $y=2t$ for $0\le t\le 5$. That is, compute ${\int }_C^{}xyds$.

2. Evaluate the definite integral from $1$ to $e$: ${\int }_1^{e}xln\left(x\right)dx$.

Evaluate the definite integral: ${\int }_0^3(1+7x) dx$.

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

Evaluate the line integral ${\int }_C^{}xy ds$, where C is the curve given by $x=t^2$, $y=2t$, for $0\le t\le 1$.

Evaluate the integral by interpreting it in terms of areas: ${\int }_0^9(3x-2) dx$.

Calculate the value of the iterated integral: ${\int }_0^3{\int }_0^1\frac{4xy}{x^2+y^2} dy dx$

Evaluate the line integral of the vector field $F(x,y)=(x+9y,x^2)$ along the curve $C$, where $C$ is the line segment from $(0,0)$ to $(1,2)$:${\int }_C^{}(x+9y)dx+x^{2}dy$

Given the curve $y=x^2$ from $x=0$ to $x=2$, find the exact length of the curve.