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Lecture5 - Lecture 5 MATH 138-W12-003 I Trigonometric...

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Lecture 5: MATH 138-W12-003 January 13, 2012 I) Trigonometric substitution, Section 7.3 In this lecture we focus on computing integrals involving functions that include a 2 - x 2 , x 2 - a 2 and x 2 + a 2 using trigonometric substitutions. For example the first occurs when we want to compute the area of an ellipse. Example If the equation of an ellipse is x 2 /a 2 + y 2 /b 2 = 1 then we can write the are as, A = 4 Z a 0 b a a 2 - x 2 dx. At the moment we do not have the techniques for evaluating this. The new technique that we be- gin to develop is to make a trigonometric substitution of the form, x = a sin θ and dx = a cos θdθ . The textbook calls this an inverse substitution because we are making this more complicated but it is still a substitution so that’s the terminology I’m going to use. There is a geometric interpretation of this substitution. We are defining a triangle that has one side x , another a 2 - x 2 and a hypotenuse of a . We can assume that the angle we are dealing with is in the range - π/ 2 θ π/ 2 . If x is positive than so is the angle but if x is negative, then the angle is negative. It is important to recall that cos θ is not negative in this interval so that | cos θ | = cos θ .
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