8. Definite Integrals

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 $0\le t\le 1$.

Write the two definite integrals subtracted below as a single integral.

${\int }_1^6\sqrt{x^2-5x}dx-{\int }_{10}^6\sqrt{x^2-5x}dx$

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.

Evaluate the definite integral: ${\int }_0^{\frac{\pi }{6}}\sin \left(t\right){\cos }^2\left(t\right) dt$.

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

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 double integral ${\int }_D^{}(2x+y)da$, where $D=\left\{\right(x,y)\mid 1\le y\le 4,y-3\le x\le 3\}$.

Let C be the curve parameterized by $x=t^2$, $y=t^3$ for $0\le t\le 1$. Find the value of the line integral ${\int }_C^{}ydx$.