Section 15.1-..

Change of Variables

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Eval some $\int$ region bounded by some region

$$\int {\int }_{D}x(y-1)dA,y=1-x^2,y=x^2-3$$

Essentially with this we aren’t doing any transformations, just integrating as such. No need to worry about jacobinan, just integrate after finding the bounds as per normal:

1. Find bounds
   1. Set both eq equal to each other
   2. Outer $\int$ is whatever result of step 1
   3. Inner $\int$ is from $h_1,h_2$
   4. If both h and g aren’t some function of x, then find where its bounded by (basically draw a graph girl)
2. Integrate normally

Where } D = $(x,y),|,a\le x\le b,g_1(x)\le y\le g_2(x)$ the integral is defined to be,

$$\underset{D}{\iint }f(x,y)dA=\underset{a}{\overset{b}{\int }}\underset{g_1(x)}{\overset{g_2(x)}{\int }}f(x,y)dydx$$