# How to Find the Point of Intersection Between ln x and 5-x?

• MHB
• karush
In summary, the conversation involves finding the point of intersection between two functions, $\ln x$ and $5-x$, which is calculated to be $x=3.69344$. The conversation then moves on to discussing how to set up integrals to find the volume and area bounded by these functions. There is also a discussion on changing variables and solving for the constant $k$.
karush
Gold Member
MHB
i

Ok this might take a while...
but first find point of intersection $\ln x=5-x$
which calculates to $x=3.69344$ which maybe there is more simpler approach

this is a calculator active problem ...

$\displaystyle R = \int_1^a \ln{x} \, dx + \int_a^5 5-x \, dx$
where $a$ is the x-value of the intersection.

or ...
$\displaystyle R = \int_0^b (5-y) - e^y \, dy$
where $b$ is the y-value of the intersection.

can you set up the volume by similar cross-section integral ?

$\displaystyle V = \int_1^a (\ln{x})^2\, dx + \int_a^5 (5-x)^2 \, dx$

ok ... continue with part (c)

skeeter said:
ok ... continue with part (c)
if we chanhge b to k
$\displaystyle \int_0^k (5-y) - e^y \, dy = \dfrac{1}{2} A$
then solve for k y was derived previous
anyway...

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