Lecture 6 - MATH 118, LECTURES 5 & 6:...

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Unformatted text preview: MATH 118, LECTURES 5 & 6: COMPLETING THE SQUARE 1 Completing the Square So far we have learned how to handle integrals with trouble terms of the form a 2- b 2 x 2 , a 2 + b 2 x 2 , and b 2 x 2- a 2 by using trigonometric substitutions. The common thread with these terms was that there was a constant term ( a 2 ) and a second-order term ( b 2 x 2 ) (i.e. x to the second power). Which substitution we used depended on which of these terms had a positive or negative sign. We might ask whether we can also handle terms with a first-order term, i.e. things like ax 2 + bx + c . It turns out the apparent problem is just a matter of representation. If we were asked, for instance, to graph the parabola y = ax 2 + bx + c , we would immediately rearrange the expression into the form y = m ( x- p ) 2 + q with new constants m , p and q . From this form, we could immediately determine the vertex of the parabola and would know whether the parabola opened up or down. The important thing to notice for our purposes is that m ( x- p ) 2 + q looks an awful lot like the troublesome terms we were solving for earlier this week in that there is one constant term q and one second-order term m ( x- p ) 2 . In fact, we can solve integrals containing troublesome terms....
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Lecture 6 - MATH 118, LECTURES 5 & 6:...

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