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Order of Integration

web.ma.utexas.edu · 1,332 words · saved by 1 readers

Some regions can be viewed either as Type I or Type II. In that case we can set up an iterated integral in two ways. Depending on the integrand, one can be a lot easier than the other! Sometimes you're given an impossible-looking iterated integral, and you can solve it by swapping (aka reversing) the order of integration. This means An example is worked in detail in the video. Example 1: Evaluate the iterated integral I = ∫ 6 0 ( ∫ 2 x/3 x 1+ y 3 − − − − − √ dy)dx. 𝐼 = ∫ 0 6 ( ∫ 𝑥 / 3 2 𝑥 1 + 𝑦 3 𝑑 𝑦 ) 𝑑 𝑥 . Solution: The inner integral is hopeless, and nothing you have learned so far in calculus will help. Instead, we need to swap the order of integration. The region of integration is the blue triangle shown on the left, bounded below by the line y= x 3 𝑦 = 𝑥 3 and above by y=2 𝑦 = 2 , since we are integrating y 𝑦 along the red line from y= x 3 𝑦 = 𝑥 3 to y=2 𝑦 = 2 . Since we are integrating x 𝑥 from 0 to 6, the left edge of the triangle is at x=0 𝑥 =

Order of Integration Home Integration by Parts Integration by Parts Examples Integration by Parts with a definite integral Going in Circles Tricks of the Trade Integrals of Trig Functions Antiderivatives of Basic Trigonometric Functions Product of Sines and Cosines (mixed even and odd powers or only odd powers) Product of Sines and Cosines (only even powers) Product of Secants and Tangents Other Cases Trig Substitutions How Trig Substitution Works Summary of trig substitution options Examples Completing the Square Partial Fractions Introduction to Partial Fractions Linear Factors Irreducible Q

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