The area of the region enclosed by the curves | vedclass

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Question

(B) Given curves are $y = x$,$y = \frac{1}{x}$,$x = e$ and the positive $X$-axis $(y = 0)$.The intersection point of $y = x$ and $y = \frac{1}{x}$ is

Vedclass · Accepted Answer

$\frac{3}{2}$

Vedclass · Answer

(B) Given curves are $y = x$,$y = \frac{1}{x}$,$x = e$ and the positive $X$-axis $(y = 0)$.The intersection point of $y = x$ and $y = \frac{1}{x}$ is found by setting $x = \frac{1}{x}$,which gives $x^2 = 1$. Since we are in the first quadrant,$x = 1$. Thus,the point of intersection is $(1, 1)$.The region is bounded by $y = x$ from $x = 0$ to $x = 1$,and by $y = \frac{1}{x}$ from $x = 1$ to $x = e$.Required Area = $\int_{0}^{1} x \, dx + \int_{1}^{e} \frac{1}{x} \, dx$$= \left[ \frac{x^2}{2} ight]_{0}^{1} + [\ln |x|]_{1}^{e}$$= \left( \frac{1}{2} - 0 ight) + (\ln e - \ln 1)$$= \frac{1}{2} + (1 - 0) = \frac{1}{2} + 1 = \frac{3}{2} 	ext{ sq units}$.

The area of the region enclosed by the curves $y = x$,$y = \frac{1}{x}$,$x = e$ and the positive $X$-axis is

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