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Find the area under the given curves and given lines: (i) $y=x^{2}$, $x=1$, $x-2$ and x-axis (ii) $y=x^{2}$,$x=1$, $x=5$ and x-axis.
Find the area under the given curves and given lines: (i) $y=x^{2}$, $x=1$, $x-2$ and x-axis (ii) $y=x^{2}$,$x=1$, $x=5$ and x-axis.
Applications of Integrals | CBSE | Class 12 | Maths | Miscellaneous | NCERT
Find the area under the given curves and given lines: (i) $y=x^{2}$, $x=1$, $x-2$ and x-axis (ii) $y=x^{2}$,$x=1$, $x=5$ and x-axis.

Solution:
(i)The area required is represented by the shaded area ADCBA as

$\begin{aligned} \text { Area } \mathrm{ADCBA} &=\int^{2} y d x \\ &=\int^{2} x^{2} d x \\ &=\left[\frac{x^{3}}{3}\right]_{1}^{2} \\ &=\frac{8}{3}-\frac{1}{3} \\ &=\frac{7}{3} \text { units } \end{aligned}$
(ii). The area required is represented by the shaded area ADCBA as


$\text { Area } \mathrm{ADCBA}=\int_{1}^{b} x^{4} d x$
$\begin{array}{l}
=\left[\frac{x^{5}}{5}\right]_{1}^{5} \\
=\frac{(5)^{5}}{5}-\frac{1}{5} \\
=(5)^{4}-\frac{1}{5} \\
=625-\frac{1}{5} \\
=624.8 \text { units }
\end{array}$

i

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