Changing Bounds for Integrals with the Jacobian

Changing Variables in Integrals §

$$\begin{array}{c}{\int }_a^{b}f(x)dx\\ x=g(u)⟹dx=g^{\prime }(u)du\\ \int f(g(u))g^{\prime }(u)du\end{array}$$

Definition of Jacobian: §

$$\begin{array}{c}\iint f(x,y)dxdy\end{array}$$

If $x=g(u,v)$ and $y=h(u,v)$ then the Jacobian of $x,y$ with respect to $u,v$ is

$$\begin{array}{c}J=\frac{\partial (x,y)}{\partial (u,v)}=\left(\begin{array}{cc}\frac{\partial x}{\partial u}& \frac{\partial y}{\partial v}\\ \frac{\partial y}{\partial u}& \frac{\partial y}{\partial v}\end{array}\right)\end{array}$$

Rectangular transformation §

$$\begin{array}{c}\iint f(x,y)dA\to \iint f(g(u,v),h(u,v))|J|dudv\end{array}$$

Polar Transformation §

$$\begin{array}{c}x=r\cos \theta ,y=r\sin \theta \\ J=\left(\begin{array}{cc}\frac{\partial x}{\partial r}& \frac{\partial x}{\partial \theta }\\ \frac{\partial y}{\partial r}& \frac{\partial y}{\partial \theta }\end{array}\right)=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)\end{array}$$

Basically converting Triple Integral for Volume with Cartesian Coordinates with Triple Integral with Cylindrical and Spherical Coordinates!

Examples §

Example 1 §

Find the Jacobian and use teh transformation to evaluate ${\iint }_R{x}^2-y^{2}dxdy$ where R is the region bounded by $y=-x+2,y=x-4,y=x+4,y=-x+4$ given $u=x+y,v=x-y$

Change region: §

$$\begin{array}{c}y=-x+2⟹x+y=2⟹u=2\\ y=-x-4⟹x+y=4⟹u=-4\\ y=x-4⟹x-y=4⟹v=4\\ y=x+4⟹x-y=-4⟹v=-4\\ f(x,y)=x^2-y^2⟹f(u,v)=uv\\ u=x+y,v=x-y⟹x=\frac{u+v}{2},y=\frac{u-v}{2}\\ J=\left(\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& -\frac{1}{2}\end{array}\right)⟹|J|=\frac{1}{2}\end{array}$$

Area: §

$$\begin{array}{rl}A& ={\int }_{u=-4}^{u=2}{\int }_{v=-4}^{v=4}u\cdot v\cdot \frac{1}{2}dvdu\end{array}$$

Example 2 §

FInd $\iint \sqrt{1-\frac{x^2}{4}-\frac{y^2}{9}}dA$ wehre R is the region bounded by $\frac{x^2}{4}+\frac{y^2}{9}=1$

Change Region §

$$\begin{array}{c}u=\frac{x}{2},v=\frac{y}{3}⟹u^2+v^2=1\end{array}$$

Jacobian §

$$\begin{array}{c}J=\left(\begin{array}{cc}2& 0\\ 0& 3\end{array}\right)⟹|J|=6\end{array}$$

Volume §

$$\begin{array}{c}{\int }_{\theta =0}^{\theta =2\pi }{\int }_{r=0}^{r=1}\sqrt{1-r^2}6\cdot rdrd\theta \end{array}$$