I. Estimating Area Under a Curve Suppose we want to estimate the area under the curve f(x) = x² +1 between x = 0 and x = 4. ту 16 12 8 .5 1.0 1.5 2.0 2.5 3.0 3.5 To find the shaded area, we will use Riemann rectangles (named for Bernard Riemann). Suppose we draw 4 rectangles to estimate the area under the curve using a left-hand sum. We break up our horizontal distance into 4 equal lengths. We take the total width, 4 - 0 = 4, and divide it by the number of rectangles, 4, to get equal-width rectangles. Thus, each rectangle is width 1. For the height of each rectangle, we'll use the left- hand endpoint (the height or y-value of the curve at the left side of the rectangle.) 4-0 4 = 5. Consider Questions 1-2 presented an increasing function. Let A be the area under the graph of an increasing continuous function f from a to b, and let L and R be the approximations to A with n subintervals using left and right endpoints, respectively. Fill in the compound inequality, putting A, LÄ, and R each in one blank. Now fill in each blank with either Ln or Rn Thus, overestimates the area A and underestimates the area A. 7

I. Estimating Area Under a Curve Suppose we want to estimate the area under the curve f(x) = x² +1 between x = 0 and x = 4. ту 16 12 8 .5 1.0 1.5 2.0 2.5 3.0 3.5 To find the shaded area, we will use Riemann rectangles (named for Bernard Riemann). Suppose we draw 4 rectangles to estimate the area under the curve using a left-hand sum. We break up our horizontal distance into 4 equal lengths. We take the total width, 4 - 0 = 4, and divide it by the number of rectangles, 4, to get equal-width rectangles. Thus, each rectangle is width 1. For the height of each rectangle, we'll use the left- hand endpoint (the height or y-value of the curve at the left side of the rectangle.) 4-0 4 = 5. Consider Questions 1-2 presented an increasing function. Let A be the area under the graph of an increasing continuous function f from a to b, and let L and R be the approximations to A with n subintervals using left and right endpoints, respectively. Fill in the compound inequality, putting A, LÄ, and R each in one blank. Now fill in each blank with either Ln or Rn Thus, overestimates the area A and underestimates the area A. 7

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 94E

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