AP EAMCET · Maths · Area Under Curves
\(O A B C\) is a unit square where \(O\) is the origin and \(B=(1,1)\). The curves \(y^2=x\) and \(x^2=y\) divide the area of the square into three parts \(O A B O, O B O\) and \(O B C O\). If \(a_1, a_2, a_3\) are the areas (in sq units) of these parts respectively, then \(a_1+2 a_2+3 a_3=\)
- A 1
- B 2
- C 6
- D 64
Answer & Solution
Correct Answer
(B) 2
Step-by-step Solution
Detailed explanation
Now, \(\begin{aligned} a_2 & =\int_0^1\left(\sqrt{x}-x^2\right) d x \\ & =\left[\frac{2}{3} x^{3 / 2}-\frac{x^3}{3}\right]_0^1=\frac{2}{3}-\frac{1}{3}=\frac{1}{3}\end{aligned}\) So, \(\quad a_1+a_3+\frac{1}{3}=1\) From Eqs. (ii) and (iii), \[ a_1=a_3=\frac{1}{3}=a_2 \] So,…
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