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

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 ${\int }_C^{}xyds$.

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 $(1,2)$:${\int }_C^{}(x+9y)dx+x^{2}dy$

Express the limit $\underset{n\to \infty }{lim}\sum i=1n\frac{2}{n}(1+\frac{2i}{n}{ }^3)$ as a definite integral on the interval $[1,3]$. Which of the following is correct?

Find the area of the surface defined by $z=2\surd x^3+y^3$ over the region $0\le x\le 1$, $0\le y\le 1$.

Consider the double integral ${\int }_0^2{\int }_{y^2}^{4}f(x,y)dxdy$. Which of the following correctly expresses the integral with the order of integration reversed?

Evaluate the double integral by reversing the order of integration: ${\int }_0^2{\int }_y^2(x+y) dx dy$.