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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Area Under Curves

NDA 2022 · Mathematics · Area Under Curves

What is the area of the region (in the first quadrant) bounded by...

What is the area of the region (in the first quadrant) bounded by y=1−x2,y=xy=\sqrt{1-x^2}, y=x and y=0y=0 ?

Options

  1. Aπ4\frac{\pi}{4}
  2. Bπ6\frac{\pi}{6}
  3. Cπ8\frac{\pi}{8}
  4. Dπ12\frac{\pi}{12}

Correct answer

C. π8\frac{\pi}{8}

Step-by-step solution


⇒x2+y2=1\Rightarrow x^2+y^2=1 Required area ∫01/2xdx+∫1/211−x2dx=[x22]01/2+[x21−x2+12sin⁡−1x]1/21=π8\begin{aligned} & \int_0^{1 / \sqrt{2}} x d x+\int_{1 / \sqrt{2}}^1 \sqrt{1-x^2} d x \\ & =\left[\frac{x^2}{2}\right]_0^{1 / \sqrt{2}}+\left[\frac{x}{2} \sqrt{1-x^2}+\frac{1}{2} \sin ^{-1} x\right]_{1 / \sqrt{2}}^1=\frac{\pi}{8} \end{aligned}
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