Evaluate the definite integral: $2{\int }_{\frac{\pi }{3}}^{\frac{\pi }{3}}{csc}^2(\frac{1}{2}t)dt$.

Given the function $f\left(x\right)$ graphed below, where $f\left(x\right)$ is a straight line from $(0,0)$ to $(2,2)$, evaluate the definite integral ${\int }_0^{2}f\left(x\right)dx$.

Evaluate the line integral ${\int }_C^{}\frac{x}{y} ds$, where C is the curve given by $x=t^3$, $y=t^4$, for $1\le t\le 2$.

Evaluate the definite integral: ${\int }_1^2{w}^2\ln \left(w\right) dw$.

Given the parametric equations $x=t^2-3t$ and $y=t$ for $t\in [0,3]$, what is the area enclosed by the curve and the y-axis?

Evaluate the double integral ${\int }_0^6{\int }_0^1(6x+1+xy) dy dx$.

Evaluate the iterated integral: $\int \underset{0}{}\stackrel{2}{}\int \underset{0}{}\stackrel{y^2}{}{x}^{2}ydxdy$

Find the exact length of the curve given by $x=2t^3$, $y=t^2-2$ for $0\le t\le 5$.

Given the parametric equations $x={\sin }^2\left(t\right)$, $y=2\cos \left(t\right)$ for $0\le t\le \pi$, what is the area enclosed by the curve and the y-axis?