Homework Help is Here – Start Your Trial Now!
Math
Trigonometry
Use rectangles to estimate the area under the parabola y - 4x² from 0 to 1 (the parabolic region S illustrated in the figure below). Solution X 3-42² We first notice that the area of S must be somewhere between 0 and 4 because S is contained in a rectangle of side lengths 1 and 4, but we can certainly do better than that. Suppose we divide Sinto four strips S₂, S. Sand S by drawing the vertical lines and x- 2 as in the figure below. y y=4x² 1 4 3 4 4 We can approximate each strip by a rectangle that has the same base as the strip and whose height is the same as the right edge of the strip (see the figure below). 2 40.4) 2 5₁ 4 We see that the area A is less than R₂, so A< 1 Ⓒ In other words, the heights of these rectangles are the values of the function f(x) = 4x² at the right endpoints of the subintervals (1,4) 1 (1,4) 1 Each rectangle has width , and the heights are 4(+)², 4(+)³, 4(2)", and 4(1)². If we let R₂ be the sum of the areas of these approximating rectangles, we get the following. R₂ − — · ‹( ▬ )³² + — · ‹( ² )³ + ½ - 4(²)² + ½ · 4(1)² and Instead of using the rectangles in the figure above we could use the smaller rectangles whose heights are the values off at the left endpoints of the sub intervals. (The leftmost rectangle has collapsed because its height is 0.) The sum of the areas of these approximating rectangles is as follows. ². − — · 4(0)² + — · 4(²)³ + — · ª(-) + ² (²) ³ - We see that the area is larger than L, so we have lower and upper estimates for A.

Use rectangles to estimate the area under the parabola y - 4x² from 0 to 1 (the parabolic region S illustrated in the figure below). Solution X 3-42² We first notice that the area of S must be somewhere between 0 and 4 because S is contained in a rectangle of side lengths 1 and 4, but we can certainly do better than that. Suppose we divide Sinto four strips S₂, S. Sand S by drawing the vertical lines and x- 2 as in the figure below. y y=4x² 1 4 3 4 4 We can approximate each strip by a rectangle that has the same base as the strip and whose height is the same as the right edge of the strip (see the figure below). 2 40.4) 2 5₁ 4 We see that the area A is less than R₂, so A< 1 Ⓒ In other words, the heights of these rectangles are the values of the function f(x) = 4x² at the right endpoints of the subintervals (1,4) 1 (1,4) 1 Each rectangle has width , and the heights are 4(+)², 4(+)³, 4(2)", and 4(1)². If we let R₂ be the sum of the areas of these approximating rectangles, we get the following. R₂ − — · ‹( ▬ )³² + — · ‹( ² )³ + ½ - 4(²)² + ½ · 4(1)² and Instead of using the rectangles in the figure above we could use the smaller rectangles whose heights are the values off at the left endpoints of the sub intervals. (The leftmost rectangle has collapsed because its height is 0.) The sum of the areas of these approximating rectangles is as follows. ². − — · 4(0)² + — · 4(²)³ + — · ª(-) + ² (²) ³ - We see that the area is larger than L, so we have lower and upper estimates for A.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 38E
See similar textbooks
icon
Related questions
Question
Use rectangles to estimate the area under the parabola y - 4x² from 0 to 1 (the parabolic region 5 illustrated in the figure below). Solution X 3-42² We first notice that the area of S must be somewhere between 0 and 4 because S is contained in a rectangle of side lengths 1 and 4, but we can certainly do better than that. Suppose we divide Sinto four strips S₂, S. Sand S by drawing the vertical lines and x- 2 as in the figure below. y y=4x² 1 4 3 4 4 We can approximate each strip by a rectangle that has the same base as the strip and whose height is the same as the right edge of the strip (see the figure below). 2 40.4) 2 5₁ 4 We see that the area A is less than R₂, so A< 1 Ⓒ In other words, the heights of these rectangles are the values of the function f(x)- 4x² at the right endpoints of the subintervals (1,4) 1 (1,4) 1 Each rectangle has width , and the heights are 4(+)², 4(+)³, 4(2)", and 4(1)². If we let R₂ be the sum of the areas of these approximating rectangles, we get the following. R₂ − — · ‹ ( ▬ )³² + — · ‹( ² )³ + ½ - 4(²)² + — · 4(1)² and Instead of using the rectangles in the figure above we could use the smaller rectangles whose heights are the values off at the left endpoints of the sub intervals. (The leftmost rectangle has collapsed because its height is 0.) The sum of the areas of these approximating rectangles is as follows. ². − —- · 4(0)² + — · 4( ² )² + — · ª(-) + ² (²) ³ - We see that the area is larger than L, so we have lower and upper estimates for A. <A< We can repeat this procedure with a larger number of strips. The figures below show what happens when we divide the region Sinto eight strips of equal width.
Transcribed Image Text:Use rectangles to estimate the area under the parabola y - 4x² from 0 to 1 (the parabolic region 5 illustrated in the figure below). Solution X 3-42² We first notice that the area of S must be somewhere between 0 and 4 because S is contained in a rectangle of side lengths 1 and 4, but we can certainly do better than that. Suppose we divide Sinto four strips S₂, S. Sand S by drawing the vertical lines and x- 2 as in the figure below. y y=4x² 1 4 3 4 4 We can approximate each strip by a rectangle that has the same base as the strip and whose height is the same as the right edge of the strip (see the figure below). 2 40.4) 2 5₁ 4 We see that the area A is less than R₂, so A< 1 Ⓒ In other words, the heights of these rectangles are the values of the function f(x)- 4x² at the right endpoints of the subintervals (1,4) 1 (1,4) 1 Each rectangle has width , and the heights are 4(+)², 4(+)³, 4(2)", and 4(1)². If we let R₂ be the sum of the areas of these approximating rectangles, we get the following. R₂ − — · ‹ ( ▬ )³² + — · ‹( ² )³ + ½ - 4(²)² + — · 4(1)² and Instead of using the rectangles in the figure above we could use the smaller rectangles whose heights are the values off at the left endpoints of the sub intervals. (The leftmost rectangle has collapsed because its height is 0.) The sum of the areas of these approximating rectangles is as follows. ². − —- · 4(0)² + — · 4( ² )² + — · ª(-) + ² (²) ³ - We see that the area is larger than L, so we have lower and upper estimates for A. <A< We can repeat this procedure with a larger number of strips. The figures below show what happens when we divide the region Sinto eight strips of equal width.
Expert Solution
steps

Step by step

Solved in 3 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS