Calculus Cheat Sheet.pdf

Improper integral an improper integral is an integral

Info icon This preview shows page 11. Sign up to view the full content.

View Full Document Right Arrow Icon
Improper Integral An improper integral is an integral with one or more infinite limits and/or discontinuous integrands. Integral is called convergent if the limit exists and has a finite value and divergent if the limit doesn’t exist or has infinite value. This is typically a Calc II topic. Infinite Limit 1. ( ) ( ) lim t a a t f x dx f x dx →∞ = 2. ( ) ( ) lim b b t t f x dx f x dx →−∞ = 3. ( ) ( ) ( ) c c f x dx f x dx f x dx = + provided BOTH integrals are convergent. Discontinuous Integrand 1. Discont. at a : ( ) ( ) lim b b a t t a f x dx f x dx + = 2. Discont. at b : ( ) ( ) lim b t a a t b f x dx f x dx = 3. Discontinuity at a c b < < : ( ) ( ) ( ) b c b a a c f x dx f x dx f x dx = + provided both are convergent. Comparison Test for Improper Integrals : If ( ) ( ) 0 f x g x on [ ) , a then, 1. If ( ) a f x dx conv. then ( ) a g x dx conv. 2. If ( ) a g x dx divg. then ( ) a f x dx divg. Useful fact : If 0 a > then 1 a p x dx converges if 1 p > and diverges for 1 p . Approximating Definite Integrals For given integral ( ) b a f x dx and a n (must be even for Simpson’s Rule) define b a n x = and divide [ ] , a b into n subintervals [ ] 0 1 , x x , [ ] 1 2 , x x , … , [ ] 1 , n n x x with 0 x a = and n x b = then, Midpoint Rule : ( ) ( ) ( ) ( ) * * * 1 2 b n a f x dx x f x f x f x ≈ ∆ + + + L , * i x is midpoint [ ] 1 , i i x x Trapezoid Rule : ( ) ( ) ( ) ( ) ( ) ( ) 0 1 2 1 2 2 2 2 b n n a x f x dx f x f x f x f x f x + + + + + + L Simpson’s Rule : ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 1 2 2 1 4 2 2 4 3 b n n n a x f x dx f x f x f x f x f x f x + + + + + + L
Image of page 11
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern