Math 104 Calculus 8.8 Improper Integrals. Math Yu
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1 Math 04 Calculus 8.8 Improper Integrals Math 04 - Yu
2 Improper Integrals Goal: To evaluate integrals of func?ons over infinite intervals or with an infinite discon?nuity. Method: We replace the bad endpoints with variables and take limits. Combines skills of integra?on and evalua?ng limits. Descrip2on: If the limit exists, we say the integral converges and if it fails to exist (this includes infinite limits), we say the integral diverges. Math 04 - Yu
3 Types of Improper Integrals An integral can be called improper with one or any combina?on of the following: Type I: contains infinity at upper limit lower limit or both Z Z 0 Z ln x x 2 dx +x 2 dx +x 2 dx Math 04 - Yu
4 Types of Improper Integrals Type II Infinite discon?nuity at lower limit Z 3 upper limit dx (x ) 2/3 Z 0 dx (x ) 2/3 so some value in between Z 3 0 dx (x ) 2/3 Math 04 - Yu
5 Type I Improper Integral Both have to converge to ensure that the LHS converge Math 04 - Yu
6 Z Examples. Compute e 2x dx if it exists. Z 2. Determine whether p dx converges or diverges. x 2 Math 04 - Yu
7 Z Examples 3. Evaluate if it converges. +x 6 dx x 2 Math 04 - Yu
8 Type II Integrals Both have to converge to ensure that the LHS converge Nicol Math 04 - Yu
9 4. Find Z 9 0 p x dx. Examples 5. Determine of what type is the integral it converges, evaluate the integral. Z 2 dx. Does it converge? If x4 Math 04 - Yu
10 Examples In general, we can have mixture of Type I and Type II integrals. 6. Determine the type(s) of the integral If it converges, evaluate the integral. Z e /x 0 x 2 dx. Does it converge? Math 04 - Yu
11 Improper Integral of /x^p For any c>0, Z c dx converges for p>, diverges for p apple. xp Z c 0 dx converges for p<, diverges for p. xp Math 04 - Yu
12 Direct Comparison WARNING: Do NOT switch the conditions and the implications. Math 04 - Yu
13 Z Examples 2 + sin x 7. Does converge or diverge? x 2 dx 8. Does Z 4 0 x +3 x 2 dx converge or diverge? Math 04 - Yu
14 Limit Comparison i.e., f g, f is of the same order as g Math 04 - Yu
15 Z Examples x 9. Does p converge or diverge? 2 x4 +5 dx Math 04 - Yu
16 Comparison Tests for Type II Comparison Tests also work for improper integral of the second type by replacing in the limit with the point where the func?on goes to infinity. 0. Does Z /2 0 cot(x)dx converge or diverge? Math 04 - Yu
17 p- test for type I integral In many cases, we can compare the integrand f(x) with func?ons of the form /x^p. Type I integral: c>0. Z c f(x)dx, wheref(x) is a nonnegative function and If f(x) when x!, then the integral converges if p>, xp diverges if p apple. If f(x) apple C x p for some constant C>0and p>, when x is large enough, then the integral converges. If f(x) C x p for some constant C>0and p<, when x is large enough, then the integral diverges. Math 04 - Yu
18 Type II integral: p- test for type II integral Z b f(x)dx, wheref(x) is a nonnegative function continuous over the interval (a, b] and If f(x) a (x a) p when x! a+, lim f(x) =, x!a + i.e., lim x!a + f(x) (x a) p = L where L is a finite positive number, then the integral converges if p<, diverges if p. C If f(x) apple for some constant C>0and p<, when x is (x a) p close to a, then the integral converges. C If f(x) for some constant C>0and p, when x is (x a) p close to a, then the integral diverges. Math 04 - Yu
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Root test Root test n Consider the limit L = lim n a n, suppose it exists. L < 1 a n is absolutely convergent (thus convergent); L > 1 (including L = ) a n is divergent L = 1 the test is inconclusive.
Power Series. Part 1. J. Gonzalez-Zugasti, University of Massachusetts - Lowell
Power Series Part 1 1 Power Series Suppose x is a variable and c k & a are constants. A power series about x = 0 is c k x k A power series about x = a is c k x a k a = center of the power series c k =
5.9 Representations of Functions as a Power Series
5.9 Representations of Functions as a Power Series Example 5.58. The following geometric series x n + x + x 2 + x 3 + x 4 +... will converge when < x
F (x) is an antiderivative of f(x) if F (x) = f(x). Lets find an antiderivative of f(x) = x. We know that d. Any ideas?
Math 24 - Calculus for Management and Social Science Antiderivatives and the Indefinite Integral: Notes So far we have studied the slope of a curve at a point and its applications. This is one of the fundamental