8. Definite Integrals

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 $0\le x\le 1$, $0\le y\le 1$.

Given the function $f\left(x\right)$ graphed below, where $f\left(x\right)$ is a straight line from $(0,0)$ to $(2,2)$, evaluate the definite integral ${\int }_0^{2}f\left(x\right)dx$.

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

Evaluate the iterated integral: $\int \underset{0}{}\stackrel{2}{}\int \underset{0}{}\stackrel{y^2}{}{x}^{2}ydxdy$

Let $g\left(x\right)={\int }_0^{x}f\left(t\right)dt$, where $f$ is the function whose graph is shown. Which of the following statements is true about $g\left(x\right)$?