Chapter 5: Q.4 (page 432)

Find the coordinates of the centroid of the region between the graph of $f (x) = x^{2}$ and the x-axis on [0, 3].

Short Answer

Expert verified

The points are $(\frac{9}{4}, \frac{9}{2})$ .

Step by step solution

Step 1. Given information.

Consider the given information,

$f 9 x) = x^{2} and [0, 3]$

Step 2. Find the centroid coordinate.

Use the given formulas to find the point.

$\begin{array}{rcl} \bar{x} & = & \frac{\int_{a}^{b} x f (x) d x}{\int_{a}^{b} f (x) d x} \\ = & \frac{\int_{0}^{3} x \cdot x^{2} d x}{\int_{0}^{3} x^{2} d x} \\ = & \frac{\int_{0}^{3} x^{3} d x}{\int_{0}^{3} x^{2} d x} \\ = & \frac{{[\frac{x^{4}}{4}]}_{0}^{3}}{{[\frac{x^{3}}{3}]}_{0}^{3}} \\ = & \frac{\frac{81}{4}}{9} \\ = & \frac{9}{4} \end{array}$

And,

$\begin{array}{l} \bar{y} = \frac{1}{2} \frac{\int_{a}^{b} {(f (x))}^{2} d x}{\int_{a}^{b} f (x) d x} \\ = \frac{1}{2} \frac{\int_{0}^{3} {(x^{2})}^{2} d x}{\int_{0}^{3} x^{2} d x} \\ = \frac{1}{2} \frac{{[\frac{x^{5}}{5}]}_{0}^{3}}{{[\frac{x^{3}}{3}]}_{0}^{3}} \\ = \frac{1}{2} \frac{\frac{3^{5}}{3}}{\frac{3^{3}}{3}} \\ = \frac{9}{2} \end{array}$

The centroid coordinate is $(\frac{9}{4}, \frac{9}{2})$ .

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Find the coordinates of the centroid of the region between the graph of $f (x) = x^{2}$ and the x-axis on [0, 3].

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the centroid coordinate.

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Find the coordinates of the centroid of the region between the graph of f(x)=x2and the x-axis on [0, 3].

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the centroid coordinate.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Logic and Functions

Geometry

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

Find the coordinates of the centroid of the region between the graph of $f (x) = x^{2}$ and the x-axis on [0, 3].