Chapter 6: Q. 15 (page 575)

The centroid of the region between the graphs of $f (x) = \sqrt{x}$ and $g (x) = 4 - x$ on $[0, 4]$

Short Answer

Expert verified

The centroid of the region between the two graphs $f (x) = \sqrt{x}$ and $g (x) = 4 - x$ in the intervals $[0, 4]$ is $[4, \frac{172}{25}]$

Step by step solution

Step 1. Given information

Given is $f (x) = \sqrt{x}$ and $g (x) = 4 - x$ in the interval $[0, 4]$

Step 2. Calculate the centroid

$(\bar{x}, \bar{y}) = \frac{\int_{a}^{b} x |f (x) - g (x)| d x}{\int_{a}^{b} |f (x) - g (x)| d x}, \frac{\frac{1}{2} \int_{a}^{b} |f (x) - g {(x)}^{2}| d x}{\int_{a}^{b} |f (x) - g (x)| d x} \int_{0}^{4} [x^{2} - (4 - x)] d x = \int_{0}^{4} (x^{2} + x - 4) d x = {[\frac{x^{3}}{3} + \frac{x^{2}}{2} - 4 x]}_{0}^{4} = \frac{40}{3} \int_{0}^{4} x [x^{2} - (4 - x)] d x = \int_{0}^{4} (x^{3} + x^{2} - 4 x) d x = {[\frac{x^{4}}{4} + \frac{x^{3}}{3} - 2 x^{2}]}_{0}^{4} = \frac{160}{3} \frac{1}{2} \int_{0}^{4} {[x^{2} - (4 - x)]}^{2} d x = \frac{1}{2} \int_{0}^{4} [x^{4} + x^{2} + 16 + 2 x^{3} - 8 x^{2} - 8 x] d x = \frac{1}{2} \int_{0}^{4} (x^{4} + 2 x^{3} - 7 x^{2} - 8 x + 16) d x = \frac{1}{2} {[\frac{x^{5}}{5} + \frac{x^{4}}{4} - \frac{7 x^{3}}{3} - 4 x^{2} + 16 x]}_{0}^{4} = \frac{1376}{15} \int_{0}^{4} [x^{2} - (4 - x)] d x = \frac{40}{3}, \int_{0}^{4} x [x^{2} - (4 - x)] d x = \frac{160}{3} \frac{1}{2} \int_{0}^{4} [x^{2} - (4 - x)] d x = \frac{1376}{15}, i n e q u a t i o n a s, (\bar{x}, \bar{y}) = \frac{\frac{160}{3}}{\frac{40}{3}}, \frac{\frac{1376}{15}}{\frac{40}{3}} = [4, \frac{172}{25}]$

Step 3. The solution

The centroid of the region between the two graph $f (x) = \sqrt{x}$ and $g (x) = 4 - x$ in the interval $[0, 4]$ is $[4, \frac{172}{25}]$

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The centroid of the region between the graphs of $f (x) = \sqrt{x}$ and $g (x) = 4 - x$ on $[0, 4]$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculate the centroid

Step 3. The solution

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The centroid of the region between the graphs offx=xandgx=4-xon0,4

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculate the centroid

Step 3. The solution

One App. One Place for Learning.

Most popular questions from this chapter

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The centroid of the region between the graphs of $f (x) = \sqrt{x}$ and $g (x) = 4 - x$ on $[0, 4]$