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NDA 2013
Exam
NDA
Year
2013
Subject
Mathematics
Chapter
Area Under Curves

NDA 2013 · Mathematics · Area Under Curves

What is the area bounded by y= x, y=0 and x=(π)/(4) ?

What is the area bounded by y=tan⁡x,y=0y=\tan x, y=0 and x=π4x=\frac{\pi}{4} ?

Options

  1. A\ln 2 square units
  2. Bln⁡22\frac{\ln 2}{2} square units
  3. C2(ln⁡2)2(\ln 2) square units
  4. DNone of the above

Correct answer

B. ln⁡22\frac{\ln 2}{2} square units

Step-by-step solution


Required area is the shadedportionOAB.shaded portion OAB.=∫0π4tan⁡(x)⋅dx=[ln⁡(sec⁡(x))]0π/4=ln⁡sec⁡(π4)−ln⁡sec⁡(0)=ln⁡(2)−ln⁡(1)=12ln⁡(2)−0=ln⁡(2)2\begin{aligned} & =\int_0^{\frac{\pi}{4}} \tan (x) \cdot d x=[\ln (\sec (x))]_0^{\pi / 4} \\ & =\ln \sec \left(\frac{\pi}{4}\right)-\ln \sec (0) \\ & =\ln (\sqrt{2})-\ln (1) \\ & =\frac{1}{2} \ln (2)-0=\frac{\ln (2)}{2} \end{aligned}$$
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