8. Definite Integrals

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 following definite integral.

${\int }_{100}^{100}\frac{\sin \left(2x\right)-100x^2}{\sqrt{x^5+\tan (3x-6)}}dx$

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 vector field $F(x,y)=(y,x)$ along the curve $C$, which is the line segment from $(0,0)$ to $(1,1)$.

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

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

Find ${\int }_0^{6}f\left(x\right)dx$ given the graph of $y=f\left(x\right)$.

Given the following definite integral of the function $f\left(x\right)=3x^2-2x$, write the simplified integral:

$-\(\int\)_4^0\!f\(\left\)(x\(\right\))\,dx$