Question

What is the area of the parabola $x^2=y$ bounded by the line $y=1$?

• (A) $\frac{1}{3}$ square unit
• (B) $\frac{2}{3}$ square unit
• (C) $\frac{4}{3}$ square units
• (D) $2$ square units

Answer 1

The Correct Option is C

Explanation:
Concept:The area under the curve $y=f(x)$ between $x=a$ and $x=b$, is given by:$\text{ Area }={\int }_a^{b}ydx$

Calculation:Here,$x^2=y$ and line $y=1$ cut the parabola$\therefore x^2=1$$\Rightarrow x=1$ and $-1$

Area $={\int }_{-1}^{1}ydx$Here, the area is symmetric about the $y$-axis, we can find the area on one side and then multiply it by $2$, we will get the area,Area ${}_1={\int }_0^{1}ydx$Area ${}_1={\int }_0^1{x}^{2}dx$$={\left[\frac{x^3}{3}\right]}_0^1=\frac{1}{3}$This area is between $y=x^2$ and the positive $x$-axis.To get the area of the shaded region, we have to subtract this area from the area of square i.e.$(1\times 1)-\frac{1}{3}=\frac{2}{3}$Total Area $=2\times \frac{2}{3}=\frac{4}{3}$ square units.Hence, the correct option is (C).