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

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?

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

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

Evaluate the definite integral: ${\int }_1^4(t^3-t^2-4) dt$.

Evaluate the double integral ${\int }_D^{}(2x+y)da$, where $D=\left\{\right(x,y)\mid 1\le y\le 4,y-3\le x\le 3\}$.

Evaluate the definite integral: ${\int }_0^{\frac{\pi }{4}}\sin \left(t\right){\cos }^2\left(t\right) dt$.