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NDA 2020
Exam
NDA
Year
2020
Subject
Mathematics
Chapter
Area Under Curves

NDA 2020 · Mathematics · Area Under Curves

What is the area of the region enclosed between the curve y^2=2 x...

What is the area of the region enclosed between the curve y2=2xy^2=2 x and the straight line y=xy=x ?

Options

  1. A12\frac{1}{2}
  2. B1
  3. C23\frac{2}{3}
  4. D2

Correct answer

C. 23\frac{2}{3}

Step-by-step solution

Two given graphs are y2=2xandy=xy^2=2 x and y=x. Point of intersections are : x2=2x⇒x2−2x=0x(x−2)=0x=0 and  2y=0 and 2\begin{aligned} & x^2=2 x \Rightarrow x^2-2 x=0 \\ & x(x-2)=0 \\ & x=0 \text{\text{ and } } 2 & y=0 \text{ and } 2 \end{aligned} So, points of intersections are (0,0)and(2,2)(0,0) and (2,2). The graph are
The required area is the shaded portion A=∣∫02(y1−y2)⋅dx∣A=\left|\int_0^2\left(y_1-y_2\right) \cdot d x\right|=∣∫02(2x−x)⋅dx∣=[22(x)3/23−x22]02=223(2)32−2=83−2=23\begin{aligned} & =\left|\int_0^2(\sqrt{2} x-x) \cdot d x\right|=\left[\frac{2 \sqrt{2}(x)^{3 / 2}}{3}-\frac{x^2}{2}\right]_0^2 \\ & =\frac{2 \sqrt{2}}{3}(2)^{\frac{3}{2}}-2=\frac{8}{3}-2=\frac{2}{3}\end{aligned}$$
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