Chapter 6: Q. 14 (page 575)

The centroid of the region between the graph of f(x) = x 2
and the x-axis on [0, 2].

Short Answer

Expert verified

The centroid of the region between the $f (x) = x^{2}$ and x-axis in the interval $(0, 2)$ $(\frac{3}{2}, \frac{2}{15})$

Step by step solution

Step 1. Given information

Given is the graph of $f (x) = x^{2}$ and the x-axis on $(0, 2)$

Step 2. Calculate the centroid of region

$f (x) = x^{2} (\bar{x}, \bar{y}) = \frac{\int_{a}^{b} x f (x) d x}{\int_{a}^{b} f (x) d x}, \frac{\frac{1}{2} \int_{a}^{b} f {(x)}^{2} d x}{\int_{a}^{b} f (x) d x} \int_{0}^{2} x^{2} d x = {[\frac{x^{3}}{3}]}_{0}^{2} \int_{0}^{2} x^{2} d x = \frac{8}{3} \int_{0}^{2} x \cdot x^{2} d x = \int_{0}^{2} x^{3} d x = {[\frac{x^{4}}{4}]}_{0}^{2} \frac{1}{2} \int_{0}^{2} {(x^{2})}^{2} d x = \frac{1}{2} \int_{0}^{2} x^{4} d x = \frac{1}{2} {[\frac{x^{5}}{5}]}_{0}^{2} = \frac{16}{5} \int_{0}^{2} x^{2} d x = \frac{8}{3}, \int_{0}^{2} x \cdot x^{2} d x = 4 a n d \frac{1}{2} \int_{0}^{2} {(x^{2})}^{2} d x = \frac{16}{5} (\bar{x}, \bar{y}) = \frac{\frac{4}{8}}{3}, \frac{\frac{16}{5}}{\frac{8}{3}} = (\frac{3}{2}, \frac{2}{15})$

Step 3. The solution

The centroid of the region between the $f (x) = x^{2}$ and x-axis in the interval $(0, 2)$ isrole="math" localid="1649141841386" $(\frac{3}{2}, \frac{2}{15})$

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The centroid of the region between the graph of f(x) = x 2
and the x-axis on [0, 2].

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculate the centroid of region

Step 3. The solution

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The centroid of the region between the graph of f(x) = x 2and the x-axis on [0, 2].

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculate the centroid of region

Step 3. The solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Applied Mathematics

Decision Maths

Probability and Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

The centroid of the region between the graph of f(x) = x 2
and the x-axis on [0, 2].