Chapter 6: Q. 31 (page 556)

In Exercises 31–34, use a weighted average over n rectangles to approximate the centroid of the region described.
The region between f(x) = √x and the x-axis on [a, b] = [1, 9], with n = 2.

Short Answer

Expert verified

$C e n t r e o f m a s s : (- 0.37, 1.22)$

Step by step solution

Step1. Given Information

region,

$f (x) = \sqrt{x}$

Step2. Calculate

$A = \int_{- 1.31}^{0.54} 2 - x^{2} - e^{x} d x A = 2 x - \frac{x^{3}}{3} - e^{x} {|_{- 1.31}}^{0.54} A = 1.45$

Step3. Continue

$\bar{x} = \frac{1}{A} \int_{a}^{b} x f (x) ρ (x) d x \Rightarrow \bar{x} = \frac{1}{1.45} \int_{- 1.31}^{0.54} (2 x - x^{3} - x e^{x}) d x \bar{x} = \frac{1}{1.45} [x^{2} - \frac{x^{4}}{4} - (x e^{x} - e^{x}) {|_{- 1.31}}^{0.54}] = - 0.3711 \bar{y} = \frac{1}{A} \int_{a}^{b} (f^{2} (x) - ρ^{2} (x)) d x \Rightarrow \frac{1}{1.45} \int_{- 1.31}^{0.54} (\frac{1}{2}) [{(2 - x)}^{2} - e^{2 x}] d x \bar{y} = 0.345 \int_{- 1.31}^{0.54} 4 - 4 x^{2} + x^{4} - e^{2 x} d x \bar{y} = 0.345 [4 x - \frac{4 x^{3}}{3} + \frac{x^{5}}{5} - \frac{e^{2 x}}{2} {|_{- 1.31}}^{0.54}] \bar{y} = 1.22 C . o . M : (- 0.37, 1.22) .$

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In Exercises 31–34, use a weighted average over n rectangles to approximate the centroid of the region described.
The region between f(x) = √x and the x-axis on [a, b] = [1, 9], with n = 2.

Short Answer

Step by step solution

Step1. Given Information

Step2. Calculate

Step3. Continue

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In Exercises 31–34, use a weighted average over n rectangles to approximate the centroid of the region described.The region between f(x) = √x and the x-axis on [a, b] = [1, 9], with n = 2.

Short Answer

Step by step solution

Step1. Given Information

Step2. Calculate

Step3. Continue

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Geometry

Calculus

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

In Exercises 31–34, use a weighted average over n rectangles to approximate the centroid of the region described.
The region between f(x) = √x and the x-axis on [a, b] = [1, 9], with n = 2.