Double Integral

Used to find volume lol

you just do it as you’d imagine lol Example

$$\begin{array}{rl}& {\int }_0^4{\int }_1^5{x}^{4}ydydx\\ & ={\int }_0^4{\left[x^4\cdot \frac{y^2}{2}\right]}_1^{5}dx\\ & =\end{array}$$

Fubini’s Theorem

If $f$ is continuous is on the rectangle $R=[a,b]\times [c,d]$, then

$$\begin{array}{r}{\iint }_{R}f(x,y)dA={\int }_a^b{\int }_c^{d}f(x,y)dydx\\ {\iint }_{R}f(x,y)dA={\int }_c^d{\int }_a^{b}f(x,y)dxdy\end{array}$$

Evaluate

$$\begin{array}{rlr}& \underset{R}{\iint }y\sin (xy)dA,& \text{where }R=[1,2]\times [0,\pi ]\\ & {\int }_0^{\pi }\underset{\text{Hold }y\text{ constant.}}{\underbrace{{\int }_1^{2}y\sin (xy)}}drdx& \\ & & \\ & {\int }_0^{\pi }{\int }_1^{2}y\sin (xy)drdx& \\ & & \\ & \text{... not doing all that}& \end{array}$$

Double Integrals over General Regions

Type 1: $R$ is bounded above and below by functions of $x$.

Type 2: $R$ is bounded on the left and right by functions of y