The area of the region bounded by the curve y | vedclass

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(C) The area $A$ of the region bounded by the curve $y = f(x)$,the $X$-axis,and the lines $x = a$ and $x = b$ is given by the integral: $A = \int_{a}^

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$\frac{1}{2} \log 2$

Vedclass · Answer

(C) The area $A$ of the region bounded by the curve $y = f(x)$,the $X$-axis,and the lines $x = a$ and $x = b$ is given by the integral: $A = \int_{a}^{b} |y| \, dx$ Here,$y = 	an x$,$a = 0$,and $b = \frac{\pi}{4}$. Since $	an x \geq 0$ in the interval $[0, \frac{\pi}{4}]$,we have: $A = \int_{0}^{\frac{\pi}{4}} 	an x \, dx$ The integral of $	an x$ is $\ln|\sec x|$: $A = [\ln|\sec x|]_{0}^{\frac{\pi}{4}}$ $A = \ln|\sec(\frac{\pi}{4})| - \ln|\sec(0)|$ $A = \ln(\sqrt{2}) - \ln(1)$ $A = \ln(2^{1/2}) - 0$ $A = \frac{1}{2} \ln 2$ Thus,the correct option is $C$.

The area of the region bounded by the curve $y = \tan x$,$x = 0$,$x = \frac{\pi}{4}$,and the $X$-axis is . . . . . . sq. units.

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