Midterm-Exam Aid

8 linear equations 95 d efinition 81 a rst order

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Unformatted text preview: ISE 4. Calculate ￿ ￿ ￿￿￿ ￿￿ ￿￿￿ ￿ ￿￿ ￿ ￿ S OLUTION. Resist the urge to factor ￿￿￿ ￿ ￿￿ ￿ ￿ and instead complete the square: ￿￿￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿￿￿ ￿ ￿￿ Make the substitution ￿ ￿ ￿￿ ￿ ￿ so that ￿￿ ￿ ￿￿, and so ￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿ Now, we are in a position to use the trigonometric substitution ￿ ￿ ￿ ￿￿￿ ￿, so that ￿￿ ￿ ￿ ￿￿￿ ￿ ￿￿￿ ￿ ￿￿, and so ￿ ￿ ￿￿ ￿ ￿ ￿￿￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿￿￿ ￿ ￿ ￿￿￿ ￿￿ ￿ ￿￿ ￿ ￿ To return to our variable ￿, we must construct the triangle generated by ￿￿￿ ￿ ￿ ￿ , yielding ￿￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿. Then ￿ ￿ ￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿￿￿ ￿ ￿ ￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿￿ ￿ ￿ where we absorb the constant ￿￿￿ ￿ ￿ into ￿ . Finally, converting back to our original variable ￿ gives ￿ ￿ ￿ ￿￿ ￿ ￿ ￿￿ ￿ ￿￿ ￿￿￿ ￿ ￿ ￿ ￿￿￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿￿￿ ￿ ￿￿ ￿ ￿ 4 Improper Integrals (7.8) Previously, we dealt with integrals on a finite interval, with continuous integrands. An improper integral is one that does not satisfy these conditions. D EFINITION 4.1. Type I improper integrals. ￿￿ (i) If ￿ ￿ ￿￿￿ ￿￿ exists for every number ￿ ￿ ￿, then ￿ ￿ ￿ ￿ ￿￿￿ ￿￿ ￿ ￿￿￿ ￿￿￿ ￿ provided this limit exists. ￿￿ (ii) If ￿ ￿ ￿￿￿ ￿￿ exists for every number ￿ ￿ ￿, then ￿ ￿ ￿ ￿￿￿ ￿￿ ￿ ￿￿￿ ￿￿ ￿￿￿￿ provided this limit exists. 16 ￿ ￿ ￿￿￿ ￿￿￿ ￿ ￿ ￿ ￿ ￿￿￿ ￿￿￿ ￿ WATERLOO SOS E XAM -AID: MATH138 M IDTERM ￿￿ ￿￿ The improper integrals ￿ ￿ ￿￿￿ ￿￿ and ￿￿ ￿ ￿￿￿ ￿￿ are called convergent if the corresponding limit exists (finite is included in this definition) and divergent if the limit does not exist. ￿￿ ￿￿ Finally, if both ￿ ￿ ￿￿￿ ￿￿ ￿￿ and ￿￿ ￿ ￿￿￿ ￿￿ are convergent, then we define ￿￿ ￿￿ ￿￿ ￿ ￿￿￿ ￿￿ ￿ ￿ ￿￿￿ ￿￿ ￿ ￿ ￿￿￿ ￿￿￿ ￿ ￿￿ ￿ Here, any real number ￿ can be used (exercise 74 in section 7.8). E XAMPLE 8. Determine if the following integral converges, and if so, find its value. ￿￿ ￿￿ ￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿￿ ￿￿ S OLUTION. By definition, if the integral converges, then ￿￿ ￿ ￿￿ ￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿￿￿￿ ￿ ￿￿ ￿ ￿￿ ￿ ￿￿￿ ￿...
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