Question

What is the area of the region bounded by the curve $f(x)=1-\frac{x^2}{4},x\in [-2,2]$ and the $x$-axis?

• (A) $\frac{8}{3}$ square units
• (B) $\frac{4}{3}$ square units
• (C) $\frac{2}{3}$ square units
• (D) $\frac{1}{3}$ square units

Answer 1

The Correct Option is A

Explanation:
We know that: The area of the region bounded by the curve $\mathrm{f}(\mathrm{x}),\mathrm{x}\in [-\mathrm{a},\mathrm{a}]$ and the $\mathrm{x}$-axis is given by:$A={\int }_{-a}^{a}f(x)dx$Therefore,The area of the region bounded by the curve $\mathrm{f}(\mathrm{x})=1-\frac{{\mathrm{x}}^2}{4},\mathrm{x}\in [-2,2]$ and the $\mathrm{x}$-axis will be:$A={\int }_{-2}^2(1-\frac{x^2}{4})dx$$\Rightarrow A={[x-\frac{x^3}{12}]}_{-2}^2$$\Rightarrow A=[2-(-2)]-[\frac{8}{12}-\frac{-8}{12}]$$\Rightarrow A=4-\frac{4}{3}$$\Rightarrow A=\frac{8}{3}$Therefore, the area of the region bounded by the curve $f(x)=1-\frac{x^2}{4},x\in [-2,2]$ and the $x$-axis is $\frac{8}{3}$ square units.Hence, the correct option is (A).