The area bounded by the curve y = ln(x) and t | vedclass

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(D) To find the area bounded by the curve $y = \ln(x)$ and the lines $y = 0$,$y = \ln(3)$,and $x = 0$,it is easier to integrate with respect to $y$.Gi

Vedclass · Accepted Answer

$2$

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(D) To find the area bounded by the curve $y = \ln(x)$ and the lines $y = 0$,$y = \ln(3)$,and $x = 0$,it is easier to integrate with respect to $y$.Given $y = \ln(x)$,we have $x = e^y$.The region is bounded by $y = 0$ (the $x$-axis) and $y = \ln(3)$,and the curve $x = e^y$ from $x = 0$ to $x = 3$.The area $A$ is given by the integral of $x$ with respect to $y$ from $y = 0$ to $y = \ln(3)$:$A = \int_{0}^{\ln(3)} e^y \, dy$Evaluating the integral:$A = [e^y]_{0}^{\ln(3)}$$A = e^{\ln(3)} - e^0$$A = 3 - 1$$A = 2$Thus,the area is $2$.

The area bounded by the curve $y = \ln(x)$ and the lines $y = 0$,$y = \ln(3)$,and $x = 0$ is equal to

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