Improper Integrals - W2020.pdf - Techniques of Integration MAT1322 C Winter 2020 Departement of Mathematics and Statistics University of Ottawa(UofO

Improper Integrals - W2020.pdf - Techniques of Integration...

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Techniques of Integration MAT1322 C Winter 2020 Departement of Mathematics and Statistics University of Ottawa (UofO) MAT1322 C - Techniques of Integration Winter 2020 1 / 29
Plan 1 Improper Integrals Type 1: Infinite Intervals Type 2: Discontinuous Integrands A Comparison Test for Improper Integrals (UofO) MAT1322 C - Techniques of Integration Winter 2020 2 / 29
Improper Integrals Type 1: Infinite Intervals Type 1: Infinite Intervals Consider the infinite region S that lies under the curve y = 1 / x 2 , above the x -axis, and to the right of the line x = 1. The area of the part of S that lies to the left of the line x = t (shaded in the figure above) is A ( t ) = Z t 1 1 x 2 dx = - 1 x t 1 = 1 - 1 t . (UofO) MAT1322 C - Techniques of Integration Winter 2020 3 / 29
Improper Integrals Type 1: Infinite Intervals We also observe that lim t →∞ A ( t ) = lim t →∞ 1 - 1 t = 1 . The area of the shaded region approaches 1 as t → ∞ So we say that the area of the infinite region S is equal to 1 and we write Z 1 1 x 2 dx = lim t →∞ Z t 1 1 x 2 dx = 1 . (UofO) MAT1322 C - Techniques of Integration Winter 2020 4 / 29
Improper IntegralsType 1: Infinite IntervalsDefinition (Definition of an Improper Integral of Type 1)(a)IfRtaf(x)dxexists for every numberta, thenZtaf(x)dxprovided this limit exists (as a finite number). MAT1322 C - Techniques of Integration Winter 2020 5 / 29
Improper Integrals Type 1: Infinite Intervals Example Determine whether the integral Z 1 1 x dx is convergent or divergent. (UofO) MAT1322 C - Techniques of Integration Winter 2020 6 / 29
Improper Integrals Type 1: Infinite Intervals Solution: According to part (a) of the above definition, we have Z 1 1 x dx = lim t →∞ Z t 1 1 x dx = lim t →∞ [ln | x | ] t 1 = lim t →∞ (ln t - ln 1) = The limit does not exist as a finite number and so the improper integral Z 1 1 x dx is divergent. (UofO) MAT1322 C - Techniques of Integration Winter 2020 7 / 29
Improper Integrals Type 1: Infinite Intervals Example Evaluate Z 0 -∞ xe x dx .

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