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 $[- \frac{π}{2}, \frac{π}{2}]$ .

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

Consider the given information.

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

Step 2. Find the centroid coordinate.

Use the given formulas to find the point.

$\begin{array}{l} \bar{x} = \frac{\int_{a}^{b} x f (x) d x}{\int_{a}^{b} f (x) d x} \\ = \frac{\int_{- π / 2}^{π / 2} x \cos x d x}{\int_{- π / 2}^{π / 2} \cos x d x} \\ = \frac{{[x \sin x - (- \cos (x))]}_{- π / 2}^{π / 2}}{{[\sin (x)]}_{- π / 2}^{π / 2}} \\ = \frac{0}{2} \\ = 0 \end{array}$

And,

$\begin{array}{l} \bar{y} = \frac{\int_{a}^{b} {(f (x))}^{2} d x}{\int_{a}^{b} f (x) d x} \\ = \frac{\int_{- π / 2}^{π / 2} \cos^{2} x d x}{\int_{- π / 2}^{π / 2} \cos x d x} \\ = \frac{{[\frac{1}{2} (x + \frac{1}{2} \sin (2 x))]}_{- π / 2}^{π / 2}}{{[\sin (x)]}_{- π / 2}^{π / 2}} \\ = \frac{\frac{π}{2}}{2} \\ = \frac{π}{4} \end{array}$

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

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 betweenf(x)=cosxand the x-axis on −π2,π2.

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

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