8. Definite Integrals

Given the graph of $f\left(x\right)$ below, evaluate the definite integral ${\int }_0^{4}f\left(x\right)dx$.

What is the average (mean) value of the function $3t^3-t^2$ over the interval $1\le t\le 2$?

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

Evaluate the line integral ${\int }_C^{}{y}^{3}ds$, where $C$ is the curve given by $x=t^3$, $y=t$, for $0\le t\le 4$.

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

Which of the following integrals represents the area of the region bounded by the curves $y=x^2$ and $y=4$ between $x=0$ and $x=2$?

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

Find the exact length of the curve $y=\frac{1}{4}{x}^2-\frac{1}{2}\ln \left(x\right)$ for $1\le x\le 2$.

Express the limit $\underset{n\to \infty }{lim}\sum i=1n\frac{2}{n}(1+\frac{2i}{n}{ }^3)$ as a definite integral on the interval $[1,3]$. Which of the following is correct?

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

For the function shown below, the definite integral from $1$ to $0$ of $f\left(x\right)$ $dx$, that is, ${\int }_0^{1}f\left(x\right)dx$, is closest to which of the following values?

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 line integral of the vector field $F(x,y)=(x+9y,x^2)$ along the curve $C$, where $C$ is the line segment from $(0,0)$ to $(1,2)$:${\int }_C^{}(x+9y)dx+x^{2}dy$

Find the exact length of the curve $y=\frac{2}{3}(1+x^2){(}^{\frac{3}{2}}$ for $0\le x\le 4$.

Given the curve $y=x^2$ from $x=0$ to $x=2$, find the exact length of the curve.