# Area of the region bounded between two curves with integration by parts

## Homework Statement

Find the area bounded between the two curves
y=34ln(x) and y=xln(x)

## Homework Equations

Integration by parts: $$\int$$udv= uv-$$\int$$vdu

## The Attempt at a Solution

First I found the intersection points of the two equation to set the upper and lower bounds. The lower bound is 0 and the upper bound is 34. My integrand is as follows--

$$\int$$34ln(x)-xln(x) with the limits of integration being from 0 to 34.

I evaluated the integral using integration by parts, and eventually came up with the following solution...

$$\int$$34ln(x)-xln(x)= 34xln(x)-34x-(x$$^{2}$$ln(x)/2) -(1/4)x$$^{2}$$

and evaluated from 0 to 34, the answer is 1156ln(34)-1156-(1156ln(34)/2)+289

I am not quite sure where my mistake is being made. I verified my answer using a graphing calculator (although that does not absolutely make my answer correct), so if anyone sees where I am making my mistake I would greatly appreciate the help!

Mark44
Mentor
When you say the lower bound is 0, I don't see how that is possible. From your integral, it appears that you are integrating with respect to x. Neither of your functions is defined at x = 0.

check this again

34xln(x)-34x-(xLaTeX Code: ^{2} ln(x)/2) -(1/4)xLaTeX Code: ^{2}

one of the sign is wrong

My mistake was in the bounds, which should have been from 1 to 34. Thank you for your help!