question_answer

The area bounded by \[f(x)={{x}^{2}},0\le x\le 1,\] \[g(x)=-x+2,1\le x\le 2\] and \[x-axis\] is

A) \[\frac{3}{2}\]

B) \[\frac{4}{3}\]

C) \[\frac{8}{3}\]

D) None of these

Correct Answer: D

Solution :

[d] Required area = Area of OAB + Area of ABC Now, Area of \[OAB=\int\limits_{0}^{1}{f(x)dx+\int\limits_{1}^{2}{g(x)dx}}\] \[=\int\limits_{0}^{1}{{{x}^{2}}dx+\int\limits_{1}^{2}{(-x+2)dx=\left. \frac{{{x}^{3}}}{3} \right|_{0}^{1}+\left[ \frac{-{{x}^{2}}}{2}+2x \right]}_{1}^{2}}\] \[=\frac{1}{3}+\left[ \left( \frac{-4}{2}+4 \right)-\left( \frac{-1}{2}+2 \right) \right]=\frac{1}{3}+\frac{1}{2}=\frac{5}{6}\] sq. unit