Integration over General Regions

Calculus 3 · Axiom Academy

LESSON Integration over General Regions A double integral over a non-rectangular region is just strips summed — vertical for Type I, horizontal for Type II — and the region tells you the limits. 1. Type I — Sweep with Vertical Strips A region is Type I (vertically simple) when it sits between two curves of x : a bottom y=g_1(x) and a top y=g_2(x) , over a fixed run . Watch a single vertical strip: its bottom rides g_1(x) and its top rides g_2(x) . The inner integral dy sums up that strip; the outer integral dx slides the strip from x=a to x=b . Inner: integrate dy up the strip, from g_1(x) to g_2(x) . Outer: sweep dx from a to b . 2. Type II — Sweep with Horizontal Strips A region is Type II (horizontally simple) when it sits between two curves of y : a left x=h_1(y) and a right x=h_2(y) , over a fixed range . Now the strip is horizontal : its left end rides h_1(y) and its right end rides h_2(y) . The inner integral dx runs along the strip; the outer integral dy slides it from y=c up to y=d . Inner: integrate dx along the strip, from h_1(y) to h_2(y) . Outer: sweep dy from c to d . 3. Same Region, Either Order, Same Number Many regions are both Type I and Type II, so you get to choose the order — and either choice must give the identical answer. Take the triangle with corners (0,0) , (2,0) , (2,2) , bounded by y=0 , x=2 , and y=x , with f(x,y)=x+y . Watch the vertical strips fill it and total up, then the same region rebuilt from horizontal strips totalling to the same value.

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