MCQ
Choose the correct answer: Area lying between the curves $y^2=4 x$ and $y=2 x$ is:
- A$\frac23$
- ✓$\frac13$
- C$\frac14$
- D$\frac34.$
✓
Answer
Correct option: B.
$\frac13$
Equation of curve (parabola) is $y^2=4 x \ldots$ (i)
$\Rightarrow\text{y}=2\sqrt{\text{x}}=2\text{x}^{\frac12}...(\text{ii})$ Equation of another curve (line) is y = 2x ...(iii) Solving eq. (i) and (iii), we get x = 0 or x = 1 and y = 0 or y = 2 Therefore, Points of intersections of circle (i) and line (ii) are O(0, 0) and A(1, 2). Now Area OBAM = Area bounded by parabola (i) and x-axis $=\Bigg|\int\limits^1_0\text{y dx}\Bigg|=\Bigg|\int\limits^1_02\text{x}^{\frac12}\text{dx}\Bigg|=$ $2\frac{\Big(\text{x}^{\frac32}\Big)^1_0}{\frac32}$ $=\frac43(1-0)=\frac43\dots(\text{iv})$ Also, Area $\Delta\text{ OAM}=$ Area bounded by parabola (iii) and x-axis $=\Bigg|\int\limits^1_0\text{y dx}\Bigg|=\Bigg|\int\limits^1_02\text{x dx}\Bigg|=2\Big(\frac{\text{x}^2}{2}\Big)^1_0$ = (1 - 0) = 1 ...(v) Now Required shaded area OBA = Area OBAM - Area of $\Delta\text{ OAM}$ $=\frac43-1=\frac{4-3}{3}=\frac13\text{ sq. units}$ Therefore, option (B) is correct.
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