8. Definite Integrals

Calculate the double integral of $f(x,y)=x+y$ over the region $R$, where $R$ is the rectangle defined by $0\le x\le 2$ and $1\le y\le 3$.

Evaluate the surface integral ${\int }_S^{}(x^{2}z+y^{2}z) dS$, where $S$ is the hemisphere $x^2+y^2+z^2=4$, $z\ge 0$.

Evaluate the iterated integral: ${\int }_0^4{\int }_0^{y^2}{x}^{2}ydxdy$

Determine the area of the region bounded by $y=x+\sin x$, the x-axis, $x=0$, and $x=\pi$.

Given the parametric equations $x=t^2-3t$ and $y=t$ for $t\in [0,3]$, what is the area enclosed by the curve and the y-axis?

Which of the following definite integrals is equal to the area under the curve $y=x^2$ from $x=0$ to $x=2$?

11-14. {Use of Tech} Compute the absolute and relative errors in using c to approximate x.

12. x = √2; c = 1.414

Evaluate the definite integral ${\int }_0^{\pi }f\left(x\right)dx$, where $f\left(x\right)=\sin \left(x\right)$ for $0\le x<\frac{\pi }{2}$ and $f\left(x\right)=2\cos \left(x\right)$ for $\frac{\pi }{2}\le x\le \pi$.

Evaluate the double integral by reversing the order of integration: ${\int }_0^2{\int }_y^2(x+y) dx dy$.

Find the area under the curve $y=x^2$ from $x=1$ to $x=3$.

Which of the following definite integrals are equal to $0$?

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 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 $e$: ${\int }_1^{e}xln\left(x\right)dx$.