8. Definite Integrals

Evaluate the definite integral: ${\int }_0^1(6x^2-6x+8) dx$.

Find the exact length of the curve given by $x=5\cos \left(t\right)-\cos \left(5t\right)$, $y=5\sin \left(t\right)-\sin \left(5t\right)$, for $0\le t\le 2\pi$.

Evaluate the definite integral: ${\int }_1^2{w}^2\ln \left(w\right) dw$.

Express the following limit as a definite integral on the interval $[0,10]$.

${\lim }_{n\to \infty }{\sum }_{k=1}^n{(x_k^{\ast }-3)}^2\Delta x$

Evaluate the double integral ${\int }_0^2{\int }_0^1(2x+1+xy) dy dx$.

Evaluate the double integral ${\int }_0^2{\int }_0^3(4-x-y)dydx$ by first identifying it as the volume of a solid. What is the value of the integral?

Given the parametric equations $x={\sin }^2\left(t\right)$, $y=8\cos \left(t\right)$ for $0\le t\le \pi$, what is the area enclosed by the curve and the y-axis?

Calculate the value of the iterated integral: ${\int }_0^3{\int }_0^1\frac{4xy}{x^2+y^2} dy dx$

Given the parametric equations $x={\sin }^2\left(t\right)$, $y=4\cos \left(t\right)$ for $0\le t\le \pi$, what is the area enclosed by the curve and the y-axis?

Evaluate the double integral of $f(x,y)=x+y$ over the region $R$ bounded by $x=0$, $x=1$, $y=0$, and $y=2$.

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

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

Which of the following limits is equal to the definite integral from $2$ to $5$ of $1+x^4$ $dx$?

Find the area of the surface defined by $z=2\surd x^3+y^3$ over the region $0\le x\le 1$, $0\le y\le 1$.