Midterm-Exam Aid

Sometimes the washer method will be easier than the

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Unformatted text preview: ￿ ￿ ￿￿￿￿￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿￿￿￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿￿￿￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿￿￿￿￿ ￿ ￿ ￿ ￿￿￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ For the second integral, we complete the sqaure, make a shift ￿ ￿ ￿ ￿ ￿ , and use the trigonometric ￿ ￿ ￿ substitution ￿ ￿ ￿￿ ￿￿￿ ￿. We have ￿￿ ￿ ￿￿ ￿￿￿￿ ￿, and we use the identity ￿￿￿￿ ￿ ￿ ￿ ￿ ￿￿￿￿ ￿. ￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿￿ ￿ ￿￿ ￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿￿￿￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿￿￿￿￿ ￿￿￿ ￿￿￿￿ ￿ ￿ ￿￿￿ ￿￿￿ ￿ ￿￿ ￿￿ ￿￿￿￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿￿￿￿ ￿ ￿￿￿ ￿ ￿ ￿￿ ￿ ￿￿￿￿ ￿ ￿ ￿￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ Since we have an even power of sine and cosine, we try to use the double angle formula, ￿￿￿￿ ￿ ￿ ￿ ￿ ￿￿￿ ￿￿ . ￿ ￿￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿￿￿ ￿￿ ￿￿ ￿ ￿ ￿ ￿￿￿ ￿￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿￿￿ ￿￿ ￿ ￿ To return to our original variable ￿, we need to construct the triangle defined by ￿ ￿ We see that so that ￿￿ ￿￿￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿￿￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿￿￿ ￿￿ ￿ ￿ ￿￿￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ 12 ￿ ￿ ￿ ￿￿￿ ￿. ￿ ￿ ￿ WATERLOO SOS E XAM -AID: MATH138 M IDTERM Then ￿ ￿￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿￿￿￿￿￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿￿￿￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿￿ ￿ ￿ ￿￿￿￿￿￿ ￿ ￿ ￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿ ￿￿￿ ￿ ￿￿ ￿ ￿ ￿￿￿￿￿￿ ￿ ￿ ￿ ￿￿￿￿ ￿ ￿ ￿ ￿￿ ￿ We have ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿￿￿￿￿ ￿ ￿￿￿￿￿￿ ￿ ￿ ￿ ￿￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿￿ ￿ ￿￿ ￿ ￿ ￿￿￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿￿￿￿￿ ￿ ￿ ￿￿ ￿￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿￿ ￿￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿￿￿￿￿￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿￿ ￿ ￿￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿ ￿￿￿￿￿￿ ￿ ￿ ￿￿￿ ￿ ￿ ￿￿ ￿ ￿￿￿￿ ￿ ￿ ￿￿￿￿ ￿ ￿ ￿￿ ￿ ￿￿￿ ￿￿￿ ￿ ￿￿￿￿￿￿￿￿￿ ￿ ￿ where ￿ is a constant. E XAMPLE 7. Calculate ￿ ￿ ￿￿ ￿￿ ￿ ￿￿￿ S OLUTION. By inspection or using the substitution ￿ ￿ ￿ ￿ ￿, we get ￿ ￿ ￿￿ ￿￿ ￿ ￿￿ ￿ ￿￿ ￿￿ ￿ ￿￿￿￿￿ ￿ ￿￿￿￿￿ ￿ ￿￿￿￿￿ ￿ ￿￿￿￿￿￿ ￿￿￿ Wait! The Fundamental Theorem of Calculus only gives this relationship between integrals and derivatives if the integrand is continuous on the closed interval ￿￿￿￿ ￿￿, which is not true here. This is a trick question: we must use the definition of improper integrals to attempt this problem, which is coming up next. E XERCISE 1. Calculate ￿ ￿ ￿￿￿￿￿...
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