From c9426e8f36b04cd18bf8dec29d871cbf676c3f3c Mon Sep 17 00:00:00 2001
From: ben-jaynes <1btjaynes@gmail.com>
Date: Mon, 5 Oct 2026 11:54:11 -0700
Subject: [PATCH] vault backup: 2026-10-05 11:54:11

---
 College/AU 26/MATH 224 (Multivar)/Class 10-5.md | 4 ++++
 1 file changed, 4 insertions(+)

diff --git a/College/AU 26/MATH 224 (Multivar)/Class 10-5.md b/College/AU 26/MATH 224 (Multivar)/Class 10-5.md
index 9b01809..3dc4fd1 100644
--- a/College/AU 26/MATH 224 (Multivar)/Class 10-5.md	
+++ b/College/AU 26/MATH 224 (Multivar)/Class 10-5.md	
@@ -12,3 +12,7 @@ This can be extended to more general domains that are not rectangles such as $[a
 Example:
 finding the volume under the function $f(x,y) = xy$ over the domain bound by 
 $$\begin{gathered} 1<x<2 \\[1.5ex] x < y < x^2 \end{gathered}$$
+We can think of finding the volume of the solid by using slices parallel to the $YZ$ plane. With the domain provided this area can be found with 
+$$ Area(x^*)=\int_{x}^{x^2}x^*ydy$$
+Since $x$ is bound by $1<x<2$ the double integral to find the volume can be written as 
+$$\begin{gather} \int_{1}^2 \left[ \int_{x}^{x^2}xy \  dy \right] dx \\[1.5ex] \int_{1}^2 \left[ \int_{x}^{x^2} \frac{xy^2}{2}   \right] dx \end{gather}$$