SATHEE: Area Question 44

Area Question 44

Question 44

Sketch the curves and identify the region bounded by $x = 1 / 2, x = 2, y = \log x$ and $y = 2^{x}$ . Find the area of this region.

(1991, 4M)

Show Answer

Solution:

The required area is the shaded portion in following figure

$∴$ The required area

$\begin{aligned} = \int_{1 / 2}^{2} (2^{x} - \log x) d x = \frac{2^{x}}{\log 2} - (x \log x - x)^{2} & = \frac{4 - \sqrt{2}}{\log 2} - \frac{5}{2} \log 2 + \frac{3}{2} sq units \end{aligned}$

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