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I encountered a problem that asks me to calculate the area between the curves $y=0$, $y=-2$, $y=log(x)$, and $x=0$.

But in order to do so, it requires to calculate and use the following integral: $\int_{-2}^{0}{e^x}{dx} = 1 - \frac{1}{e^2}$.

I can't find a way of using that value, since the logarithm function is in base 10. (If $y=ln(x)$ I could claim that there is symmetry along $x=y$ and that would be proof enough to say the area is the same, wouldn't it?).

What step should I look into?

Thanks a lot.

Johnny
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3 Answers3

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If you draw out the graphs, you will realise that the area you are trying to find is $$\int_{-2}^0 x dy= \int_{-2}^0 e^y dy$$
and that looks very similar to $\int_{-2}^{0}{e^x}{dx}$ with the only difference being that $y$ has been renamed to $x$. In other words, when you evaluate $\int_{-2}^{0}{e^x}{dx}$ and $\int_{-2}^{0}{e^y}{dy}$ you should get the same value.

Then you can use the result $\int_{-2}^{0}{e^x}{dx} = 1 - \frac{1}{e^2}$ to deduce that the area you are looking for is in fact $1 - \frac{1}{e^2}$.

EDIT: I am assuming that by $log (x)$ you are referring to the natural logarithm.

Vizuna
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Hint: what relation is there between the graph of a function and the one of its inverse?

Edit:

If $\log$ is the natural base logarithm, the two areas coincide. In case it stands for $\log_{10}$, you could use the fact that $$\log_{10}(x)=\frac{\ln(x)}{\ln(10)}$$

Lonidard
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  • If we are talking in terms of real number calculus and not about complex variables, then $logx$ is meant to be base 10 – imranfat Mar 05 '15 at 21:24
  • I thought so at first, but everywhere else it is notated as $ln$. – Johnny Mar 05 '15 at 21:27
  • @imranfat This is not true. In mathematics, $\log$ is often intended as the natural logarithm, which is the most natural way of defining it. It depends on the notation used, which is generally specified within the book. If you can clear it up, I can help you better! – Lonidard Mar 05 '15 at 21:30
  • Without context, I would have assumed that $\log(x)=\ln(x)$. That being said, the limits of integration in the hint point toward it being base 10. – rnrstopstraffic Mar 05 '15 at 21:32
  • @bharb I agree with your assertion of the most natural way of how the natural log is defined. But I truly wonder how many people in the undergrad engineering program who study real number calculus will assume $logx$ to be meant as $lnx$ where all the textbooks I have seen use $lnx$ specifically for that purpose. Only in complex analysis the notation $logx$ is assumed to be $lnx$ since that is the only log of interest. The books that cover 18 credits of calculus in the undergrad program here in North America assume $logx$ to be base 10. And that's what I honestly thought about the OP's question – imranfat Mar 05 '15 at 22:14
  • Interesting. In Italy it often indicates natural base logarithm, both in high school and university! – Lonidard Mar 05 '15 at 22:17
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$ y= \log_{10} x = \dfrac {\log x}{\log 10} $

Now find inverse exp function with this constant else what you did is correct.

Narasimham
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  • Then $x = e^{ln(10)}e^y$ would be the inverse function g(y)=x. Using the value that I have, by linearity properties of the integral, I could multiply by $e^{ln(10)}$, which is a constant. Then the area is equal to $e^{ln(10)}(1-\frac{1}{e^2})$. – Johnny Mar 05 '15 at 21:24