Calculus for Life Sciences Students
Calculus for Life Sciences Students MATH 3A
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This 1 page Class Notes was uploaded by Kaylin Wehner on Friday September 4, 2015. The Class Notes belongs to MATH 3A at University of California - Los Angeles taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/177839/math-3a-university-of-california-los-angeles in Mathematics (M) at University of California - Los Angeles.
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Date Created: 09/04/15
Math 30 Discussions 2A X 2B 1 Review Improper Integrals Integration by Parts L7Hospital7s Rule When learning about continuous distributions you will need to calculate many integrals to nd distribu tion functions expected value variance etcl Below are excerpts from Math SAB study guides that go over integration techniques commonly used in probability namely improper integrals and integration by parts Ilve also included the section on l Hospitalls rule which may come in handy when evaluating some of the improper integralsl Section 74 Improper Integrals This section covers integrals with either in nite limits of integration or an unbounded integrand Here we will only discuss the rst type To integrate a function over an in nite interval we need to use limits 0 ftdt 13100 ftdt 0 ftdt 21320 ftdt 7 NW det 0 ftdt some a e R If these limits exist we say that the integral convergesl Otherwise the integral diverges For example 00 1 idz 17 p gt 1 l 1 1p 00 0 lt p g 1 So the integral converges if p gt 1 otherwise it divergesl An important observation is that 00 z 00 f To see why this is true consider the function 1 Using the de nition ff oo diverges but using the above formula ff 0 Keep this example in mind if youlre ever tempted to use the shortcut abovel Section 72 Integration by Parts Another cool way of solving dif cult integrals is integration by parts if and vz are differentiable then uzv zdz 7 uzvzdzl An easier way of remember this formula is f udv uv 7 f vdul Just as with the substitution rule integration by parts can also be used with de nite integrals but in that case the limits of integration do not need to be changed Integration by parts is more versatile than substitution because there are several tricks that can be used with it to solve a dif cult integrall For example there are also some cases where you need to apply integration by parts multiple times then use algebra to solve for the integral Or when applying integration by parts you may have to let v 1 Advice 0 If the integral looks like sinazdz cosazdz or fPze dz where 131 is a polynomial then let u 131 and v sinazcosaz e respectively
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