Changing Order of Integration

Calculus 3 · Axiom Academy

LESSON Changing the Order of Integration The same region, described two ways — read the new limits off the picture, don't just swap the symbols. 1. One Region, Two Ways to Sweep It Fix a region R . You can fill it with vertical strips — for each x , let y run from the lower boundary g_1(x) up to g_2(x) — or with horizontal strips — for each y , let x run from the left boundary h_1(y) across to h_2(y) . Watch the same triangle get swept both ways. Vertical strips: inner dy , outer dx Horizontal strips: inner dx , outer dy 2. Read the New Limits Off the Boundary To reverse the order you must re-solve the boundary curves for the other variable. Here R is bounded by y=x , y=1 , and the y -axis. To sweep with horizontal strips, ask: at height y , where does the strip start and end? It starts on x=0 and ends on the line y=x , i.e. x=y . The outer variable y then runs over the region's full vertical extent. Outer: . Inner: y from x up to 1 . Outer: . Inner: x from 0 across to y . 3. When One Order Is the Only Order Sometimes reversing isn't just easier — it's the only way. Take f(x,y)=e^ y^2 over that same triangle. In the original order the inner integral has no elementary antiderivative — you're stuck. Reverse it and the inner integral becomes routine. Original order: inner dy is a dead end Reversed order: inner dx resolves cleanly Reversing gives , and the substitution u=y^2 finishes it:

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