22. Find the area between the curves y = x and y = x2

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Application of Integrals Ncert Solutions Maths class 12th Maths Class 12th Application of Integrals

22. Find the area between the curves y = x and y = x²

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Vishal Baghel | Contributor-Level 10

4 months ago

The given equation of the curve is $y = x^{2}$ --------(1)

and that of the line is $y = x$ ---------(2)

Solving eq (1) and (2)for x and y

$\begin{array}{l} x = x^{2} \\ = x^{2} - x = 0 \\ = x (x - 1) = 0 \\ = x = 0 & x = 1 \end{array}$

Where, $x = 0, y = 0^{2} = 0$

And when $x = 1, y = 1^{2} = 1$

$∴$ The point of intersection of the parabola $y = x^{2}$ and the line $y = x$

Is O(0,0) and B (1,1)

Hence, area between the curve and the line is

$a r e a (D C A O) = a r e a (? O A B) - a r e a (O A B O)$

$\begin{array}{l} = \int_{0}^{1} y_{l i n e} d x - \int_{0}^{1} y_{c u r v e} d x \\ = \int_{0}^{1} x d x - \int_{0}^{1} x^{2} d x \\ = {[\frac{x^{2}}{2}]}_{0}^{1} - {[\frac{x^{3}}{3}]}_{0}^{1} \end{array}$

$\begin{array}{l} = \frac{1}{2} - \frac{1}{3} \\ = \frac{3 - 2}{6} = \frac{1}{6} u n i t^{2} \end{array}$

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