StellarSometimes it is easier to describe the region using horizontal slices instead of vertical ones. In that case, you integrate with respect to y and use right minus left instead of top minus bottom. The formula becomes
because each horizontal slice has width equal to the difference in x-values and thickness dy. This approach is especially useful when the curves are naturally given as functions of y or when solving for x makes the setup cleaner.
You should choose functions of y when vertical slices would force you to split the region awkwardly or when the left and right boundaries are simpler than the top and bottom boundaries. The main idea is the same as before: area comes from adding up slice lengths times slice thickness. The only change is the orientation of the slices and the variable of integration. This flexibility is important because some regions are much easier to handle one way than the other.