Evaluate the definite integral: ${\int }_0^3(1+7x) dx$.

Given the region bounded by $y=x^2$, $x=0$, and $y=4$, determine by direct integration the x-coordinate of the centroid of the area.

Find the length of the curve given by $r\left(t\right)=(8t,t^2,\frac{t^3}{12})$ for $0\le t\le 1$.

Evaluate the surface integral ${\int }_S^{}(x^{2}z+y^{2}z) dS$, where $S$ is the hemisphere $x^2+y^2+z^2=4$, $z\ge 0$.

2. Evaluate the definite integral from $1$ to $e$: ${\int }_1^{e}xln\left(x\right)dx$.

Evaluate the line integral ${\int }_C^{}{y}^{3}ds$, where $C$ is the curve given by $x=t^3$, $y=t$, for $0\le t\le 3$.

Evaluate the line integral ${\int }_C^{}xy ds$, where C is the curve given by $x=t^2$, $y=2t$, for $0\le t\le 1$.

Evaluate the line integral of the function $f(x,y)=xy$ along the curve $C$, where $C$ is given by the parametric equations $x=t^2$, $y=2t$ for $0\le t\le 5$. That is, compute ${\int }_C^{}xyds$.

Evaluate the integral by interpreting it in terms of areas: ${\int }_0^9(3x-2) dx$.