Area Question 34
Question 34
- Find the area of the region bounded by the curves
$$ y=x^{2}, y=\left|2-x^{2}\right| \text { and } y=2 \text {, } $$
which lies to the right of the line $x=1$.
$(2002,5$ M)
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Solution:
- The points in the graph are
$\therefore$ Required area
$$ \begin{aligned} & =\int_{1}^{\sqrt{2}}\left{x^{2}-\left(2-x^{2}\right)\right} d x+\int_{\sqrt{2}}^{2}\left{2-\left(x^{2}-2\right)\right} d x \ & =\int_{1}^{\sqrt{2}}\left(2 x^{2}-2\right) d x+\int_{\sqrt{2}}^{2}\left(4-x^{2}\right) d x \ & =\frac{2 x^{3}}{3}-2 x+4 x-\frac{x^{3}}{3} \ & =\frac{4 \sqrt{2}}{3}-2 \sqrt{2}-\frac{2}{3}+2+8-\frac{8}{3}-4 \sqrt{2}+\frac{2 \sqrt{2}}{3} \ & =\frac{20-12 \sqrt{2}}{3} \text { sq units } \end{aligned} $$
