lec18 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

Info iconThis 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 .
Background image of page 1

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

View Full DocumentRight Arrow Icon
Lecture 18 18.01 Fall 2006 Lecture 18: Defnite Integrals Integrals are used to calculate cumulative totals, averages, areas. Area under a curve : (See Figure 1.) 1. Divide region into rectangles 2. Add up area of rectangles 3. Take limit as rectangles become thin a b a b (i) (ii) Figure 1: (i) Area under a curve; (ii) sum of areas under rectangles Example 1. f ( x ) = x 2 , a = 0, b arbitrary 1. Divide into n intervals Length b/n = base of rectangle 2. Heights: ± 2 b b 1 st : x = , height = n n ± 2 2 b 2 b 2 nd : x = , height = n n Sum of areas of rectangles: ± ± 2 ± ± 2 ± ± 2 ± ± 2 b b b 2 b b 3 b b nb b 3 + + + + = (1 2 + 2 2 + 3 2 + + n 2 ) n n n n n n ··· n n n 3 ··· 1
Background image of page 2
Lecture 18 18.01 Fall 2006 a=0 b 2 Figure 2: Area under f ( x ) = x above [0 ,b ]. We
Background image of page 3

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

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/18/2012 for the course MATH 18.01 taught by Professor Brubaker during the Fall '08 term at MIT.

Page1 / 6

lec18 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online