THe graph of y = f (x) passes through the origin. the arc length of the curve from (0,0) to (x, f(x)) is given by s(x) = (integral from a to x) of the square roote of 1 + e ^ t dt... Identify the function f.
Hello, 09Ashaw!
The arc length formula is: .$\displaystyle s(x) \;=\;\int^b_a\sqrt{1 + \left(\tfrac{dy}{dx}\right)^2}\,dx$The graph of $\displaystyle y \,=\, f(x)$ passes through the origin.
The arc length of the curve from $\displaystyle (0,0)$ to $\displaystyle (x, f(x))$ is given by: .$\displaystyle s(x) \:=\:\int^x_0\sqrt{1 + e^t}\,dt$
Identify the function $\displaystyle f(x)$
So we know that: .$\displaystyle \left(\frac{dy}{dx}\right)^2 \:=\:e^t \quad\Rightarrow\quad \frac{dy}{dx} \:=\:e^{\frac{x}{2}} \quad\Rightarrow\quad dy \:=\:e^{\frac{x}{2}}\,dx $
Integrate: .$\displaystyle \int dy \;=\;\int e^{\frac{x}{2}}\,dx \quad\Rightarrow\quad y \;=\;2e^{\frac{x}{2}} + C$
Since the graph contains the origin $\displaystyle (0,0)$, we have: .$\displaystyle 0 \;=\;2e^0 +C \quad\Rightarrow\quad C \:=\:-2$
. . Therefore: .$\displaystyle \boxed{f(x) \;=\;2e^{\frac{x}{2}} - 2}$