lec24 - MIT OpenCourseWare http/ocw.mit.edu 18.01 Single...

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Lecture 24 18.01 Fall 2006 Lecture 24: Numerical Integration Numerical Integration We use numerical integration to find the definite integrals of expressions that look like: b (a big mess) a We also resort to numerical integration when an integral has no elementary antiderivative. For instance, there is no formula for x 3 cos( t 2 ) dt or e x 2 dx 0 0 Numerical integration yields numbers rather than analytical expressions. We’ll talk about three techniques for numerical integration: Riemann sums, the trapezoidal rule, and Simpson’s rule. 1. Riemann Sum a b Figure 1: Riemann sum with left endpoints: ( y 0 + y 1 + . . . + y n 1 x Here, x i x i 1 = Δ x (or, x i = x i 1 + Δ x ) a = x 0 < x 1 < x 2 < . . . < x n = b y 0 = f ( x 0 ) , y 1 = f ( x 1 ) , . . . y n = f ( x n ) 1
Image of page 2
Lecture 24 18.01 Fall 2006 2. Trapezoidal Rule The trapezoidal rule divides up the area
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
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