The area of the region enclosed by the parabolas y = x 2 - 5 x and y = 7 x - x 2 is ___________

Question

The area of the region enclosed by the parabolas $y = x^{2} - 5 x$ and $y = 7 x - x^{2}$ is ___________ [2024]

Smart Achievers · Accepted Answer

(72)

Given, $y = x^{2} - 5 x$ and $y = 7 x - x^{2}$

Both curves passes through the origin

For the points of intersection, we have $x^{2} - 5 x = 7 x - x^{2}$

$\Rightarrow 2 x^{2} = 12 x$

$\Rightarrow 2 x (x - 6) = 0$

$\Rightarrow x = 0, 6$

$∴$ Area bounded by the curves $= \int_{0}^{6} [(7 x - x^{2}) - (x^{2} - 5 x)] d x$

$= \int_{0}^{6} (7 x - x^{2} - x^{2} + 5 x) d x = \int_{0}^{6} (12 x - 2 x^{2}) d x$

$= 12 (\frac{6^{2}}{2}) - \frac{2}{3} {(6)}^{3}$

$= 72 sq. units$

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