Univariate transformation/change of variable with integration

Question

I'm trying to solve the following problem:

My thought process is to do a univariate change of variable, so that $Y = \frac{V}{100,000}$. Then we have $\frac{\delta}{\delta v}[\frac{V}{100,000}] = \frac{1}{100,000}$.

I found that the constant k = 5 by integrating the pdf from 0 to 1 and setting it equal to 1.

Now the new pdf becomes:

$f_V(v)=\frac{5}{100,000}(1-\frac{v}{100,000})^4$.

So if I want to solve this, then I would want to integrate from $400,000$ to $ 100000$ on the numerator, and $10,000$ to $100000$ on the denominator?

Any help would be appreciated.

Answer 1

$\Large \frac{\int_{.1}^1 5(1-y)^4 dy}{\int_{.4}^1 5(1-y)^4 dy}$ = $\Large \frac{.07776}{.59049}$ = $.13169$