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

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

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?

Find the area of the part of the plane $5x+4y+z=20$ that lies in the first octant.

Find the area of the region enclosed by one loop of the curve $r=\sin \left(8\theta \right)$.

Evaluate the definite integral: ${\int }_0^{\frac{\pi }{6}}\sin \left(t\right){\cos }^2\left(t\right) dt$.