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College/AU 26/MATH 224 (Multivar)/Chapter 15.3 - Double Integrals in Polar Coordinates.md

@@ -5,5 +5,15 @@ $$\begin{gather}
 	a\leq r\leq b \\ \alpha \leq \theta \leq \beta
 \end{gather}$$
 
+![[PolarRegion.excalidraw]]
 
-if a function is defined as $f(x,)$
+if a function is defined as $f(x,y)$ and the region it is integrated over is polar the integral will often look similar to 
+$$\int_{\alpha}^\beta \int_{a}^b f(r\cos(\theta), r\sin(\theta))r \ dr d\theta$$
+
+
+Find the volume of the solid below $z=1-x^2-y^2$ and above the first quadrant on the xy plane
+
+rearranging the equation gives $x^2+y^2=1-z$ which shows that each horizontal slice of the function is a circle centered at $(0,0)$ with a radius of $\sqrt{ 1-z }$
+
+since the slice of the equation at $z=0$ forms the circle $x^2+y^2=1$ the bounds 
+$$$$