This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: AREAS AND DISTANCES Suppose that we want to find the area under a curve. First of all, we need to define what the area is. If we have a rectangle, it is relatively easy, because we can simply define the area as the product of the length and the width. We will use this property of a rectangle as a base point of defining the area under a curve. Example 1. Use rectangles to estimate the area under the parabola y = x 2 from 0 to 1. Solution Suppose we divide the parabolic region into four strips. We can ap proximate each strip by a rectangle whose base is the same as the strip and whose height is the same as the right edge of the strip. Each rectangle has width 1 4 and the heights are ( 1 4 ) 2 , ( 2 4 ) 2 , ( 3 4 ) 2 , and 1 2 . If we let R 4 be the sum of the areas of these rectangles, we get R 4 = 1 4 ( 1 4 ) 2 + 1 4 ( 2 4 ) 2 + 1 4 ( 3 4 ) 2 + 1 4 1 2 = 15 32 = 0 . 46875 We can also see that the area A of the parabolic region is less than R 4 , so A < . 46875 Alternatively, we can use smaller rectangle whose heights are the values of f at the endpoints of the subintervals. The sum of these approximating rectangles is L 4 = 1 4 2 + 1 4 ( 1 4 ) 2 + 1 4 (...
View
Full
Document
This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.
 Fall '09
 BenedictGross

Click to edit the document details