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 iterated integral: ${\int }_0^2{\int }_0^1{\left(x+y\right)}^{2}dxdy$

Evaluate the definite integral ${\int }_0^{\pi }f\left(x\right)dx$, where $f\left(x\right)=\sin \left(x\right)$ for $0\le x<\frac{\pi }{2}$ and $f\left(x\right)=2\cos \left(x\right)$ for $\frac{\pi }{2}\le x\le \pi$.

Evaluate the line integral ${\int }_C^{}(2x+3y) dx+(x-y) dy$, where C is the line segment from $(0,0)$ to $(2,4)$.

Determine the area under the curve $y=x+\sin \left(x\right)$ from $x=0$ to $x=\pi$.

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