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

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.

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?

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: ${\int }_0^2{\int }_y^2(x+y) dx dy$.

Evaluate the definite integral: ${\int }_0^1(6x^2-6x+8) dx$.

Evaluate the double integral ${\int }_0^2{\int }_0^3(4-x-y)dydx$ by first identifying it as the volume of a solid. What is the value of the integral?