Q.5 Find the coordinates of the cent... [FREE SOLUTION]

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Chapter 5: Q.5 (page 432)

Find the coordinates of the centroid of the region between $f (x) = \cos x$ and the x-axis on $[鈭� \frac{蟺}{2}, \frac{蟺}{2}]$ .

Short Answer

Expert verified

The coordinate point is $(0, \frac{蟺}{4})$ .

Step by step solution

Step 1. Given information.

Consider the given information.

$f (x) = \cos x and [- \frac{蟺}{2}, \frac{蟺}{2}]$

Step 2. Find the centroid coordinate.

Use the given formulas to find the point.

$\begin{array}{l} \overset{炉}{x} = \frac{{鈭�}_{a}^{b} x f (x) d x}{{鈭�}_{a}^{b} f (x) d x} \\ = \frac{{鈭�}_{- 蟺 / 2}^{蟺 / 2} x \cos x d x}{{鈭�}_{- 蟺 / 2}^{蟺 / 2} \cos x d x} \\ = \frac{{[x \sin x - (- \cos (x))]}_{- 蟺 / 2}^{蟺 / 2}}{{[\sin (x)]}_{- 蟺 / 2}^{蟺 / 2}} \\ = \frac{0}{2} \\ = 0 \end{array}$

And,

$\begin{array}{l} \overset{炉}{y} = \frac{{鈭�}_{a}^{b} {(f (x))}^{2} d x}{{鈭�}_{a}^{b} f (x) d x} \\ = \frac{{鈭�}_{- 蟺 / 2}^{蟺 / 2} \cos^{2} x d x}{{鈭�}_{- 蟺 / 2}^{蟺 / 2} \cos x d x} \\ = \frac{{[\frac{1}{2} (x + \frac{1}{2} \sin (2 x))]}_{- 蟺 / 2}^{蟺 / 2}}{{[\sin (x)]}_{- 蟺 / 2}^{蟺 / 2}} \\ = \frac{\frac{蟺}{2}}{2} \\ = \frac{蟺}{4} \end{array}$

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91影视

Find the coordinates of the centroid of the region between $f (x) = \cos x$ and the x-axis on $[鈭� \frac{蟺}{2}, \frac{蟺}{2}]$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the centroid coordinate.

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91影视

Find the coordinates of the centroid of the region betweenf(x)=cosxand the x-axis on 鈭�蟺2,蟺2.

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

Logic and Functions

Mechanics Maths

Geometry

Theoretical and Mathematical Physics

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Decision Maths

Study anywhere. Anytime. Across all devices.

Find the coordinates of the centroid of the region between $f (x) = \cos x$ and the x-axis on $[鈭� \frac{蟺}{2}, \frac{蟺}{2}]$ .