Hey guys I'm having trouble proving the area under the curve is the same as the arc length the question is:
$f(x)= \frac{1}{4}e^x+e^{-x}$ , prove that's the arc length on any interval has the same value as area under the curve.
For the arc length I got $\frac{1}{4}(e^x-4e^{-x})+C$. But I didn't get that for the area under the curve. Please help thank you !!
Hints:
Let $\displaystyle y = f(x)$
Arc length between $X_1$ and $X_2$ is given by $\displaystyle \int_{X_1}^{X_2} \sqrt{1 + (y')^2}dx$
Area under the curve between $X_1$ and $X_2$ is given by $\displaystyle \int_{X_1}^{X_2} ydx$
So all you need to do is prove that $\displaystyle \sqrt{1 + (y')^2} = y$ holds identically true.
Can you show that $\displaystyle {1 + (y')^2} = y^2$ for all $x$?