Given that
variable $\tilde{p}$ follows a probability distribution function $\Omega$ with mean $\bar{p}$ lower and upper limits $a$ and $b$, repectively: $\tilde{p}\sim\Omega(a,b)$, and
variable $\tilde{c}$ follows a probability distribution function $\Phi$ with mean $\bar{c}$ and lower and upper limits $c$ and $d$, repectively: $\tilde{c}\sim\Phi(c,d)$,
If the values of $a, b, c, d, k,\bar{c}, \bar{p}$ are known, how would I evaluate the equation
$$E[NPV]=\int_{a}^{b} \int_{c}^{k} (\tilde{p}-\tilde{c}) \ \mathrm d \Phi(\tilde{c})\mathrm d \Omega(\tilde{p})$$
where $c<k<d$ and $E[NPV]$ is the expected value of the Net Present Value (NPV) for the cash flows represented by the two variables $\tilde{p}$ and $\tilde{c}$?
I suspect this has to do with Lebesgue integrals that are used to represent expected values of probability distributions, but I am quite unsure whether I'm correct and how I would evaluate the equation above in terms of $a,b,c$ and $k$. If an evaluation is not possible given only the information above, any simplification that removes the double integrals, or expresses the differentials in terms of $\mathrm d \tilde{c}\mathrm d \tilde{p}$ (instead of $\mathrm d \Phi(\tilde{c})\mathrm d \Omega(\tilde{p})$) would also be helpful.
This equation is a shortened version of one which I found in an academic paper. Nothing is mentioned in the source regarding the joint distribution of $(\tilde{p},\tilde{c})$
EDIT 1: it seems that, by definition, $\int_{a}^{b}\tilde{p}\mathrm d \Omega(\tilde{p})=\bar{p}$ and $\int_{c}^{d}\tilde{c}\mathrm d \Phi(\tilde{c})=\bar{c}$. The question therefore I believe is how would one evaluate $\int_{c}^{k}\tilde{c}\mathrm d \Phi(\tilde{c})$ if $c<k<d$?
EDIT 2: as pointed out in the comment, since no info regarding the joint dist. of the variables are given, I think it's safe to assume for now that $E[\tilde{p}|\tilde{c}]=E[\tilde{p}]$, and $E[\tilde{c}|\tilde{p}]=E[\tilde{c}]$
EDIT 3: after looking into Lebesgue integrals my attempt at evaluating the equation above is as follows:
$$E[NPV]=\int_{a}^{b} \int_{c}^{k} (\tilde{p}-\tilde{c}) \ \mathrm d \Phi(\tilde{c})\mathrm d \Omega(\tilde{p})$$ $$=\int_{a}^{b} \int_{c}^{k} (\tilde{p}) \ \mathrm d \Phi(\tilde{c})\mathrm d \Omega(\tilde{p})-\int_{a}^{b} \int_{c}^{k} (\tilde{c}) \ \mathrm d \Phi(\tilde{c})\mathrm d \Omega(\tilde{p})$$
$$=\int_{a}^{b} \tilde{p}[\Phi(k)]\mathrm d \Omega(\tilde{p})-\int_{a}^{b} k\Phi(k) \mathrm d \Omega(\tilde{p})$$
$$=\bar{p}[\Phi(k)]-k[\Phi(k)]\text{ (outer integral of 2nd term = 1)}$$
$$=(\bar{p}-k)\cdot\Phi(k)$$
EDIT 4: As per the corrections in the comments, this is my second attempt: $$E[NPV]=\int_{a}^{b} \int_{c}^{k} (\tilde{p}-\tilde{c}) \ \mathrm d \Phi(\tilde{c})\mathrm d \Omega(\tilde{p})$$ $$=\int_{a}^{b} \int_{c}^{k} (\tilde{p}) \ \mathrm d \Phi(\tilde{c})\mathrm d \Omega(\tilde{p})-\int_{a}^{b} \int_{c}^{k} (\tilde{c}) \ \mathrm d \Phi(\tilde{c})\mathrm d \Omega(\tilde{p})$$
$$=\int_{a}^{b} \tilde{p}[\Phi(k)]\mathrm d \Omega(\tilde{p})-\int_{a}^{b} \left(k\Phi(k)-\int_{c}^{k}\Phi(\tilde{c}) \mathrm d\tilde{c}\right) \mathrm d \Omega(\tilde{p})$$
$$=\bar{p}[\Phi(k)]-k[\Phi(k)]+\Phi(k)\text{ (outer integral of 2nd and 3rd term = 1)}$$
$$=(\bar{p}-k+1)\cdot\Phi(k)$$ $$\text{where } \Phi(k)=\mathbb{P}(\tilde{c}\leq k)$$
EDIT 6: in light of additional comments, here's another attempt at the question
re-define $\phi(\tilde{c})$ and $\omega(\tilde{p})$ as the PDFs of $\tilde{c}$ and $\tilde{p}$ over the ranges $[c,d]$ and $[a,b]$, respectively, and $\Phi(\tilde{c})$ and $\Omega(\tilde{p})$ are their respective CDFs; for $c<d<k, \Phi(d)=\Omega(b)=1, \Phi(c)=\Omega(c)=0$:
$$E[NPV]=\int_{a}^{b} \int_{c}^{k} (\tilde{p}-\tilde{c}) \ \mathrm d \Phi(\tilde{c})\mathrm d \Omega(\tilde{p})$$
$$=\int_{a}^{b} \int_{c}^{k} (\tilde{p})\phi(\tilde{c})\omega(\tilde{p}) \ \mathrm d \tilde{c}\mathrm d \tilde{p}-\int_{a}^{b} \int_{c}^{k} (\tilde{c})\phi(\tilde{c})\omega(\tilde{p}) \ \mathrm d \tilde{c}\mathrm d \tilde{p}$$
$$=[\Phi(\tilde{c})]_c^k\int_{a}^{b} (\tilde{p})\omega(\tilde{p})\mathrm d (\tilde{p})- [\tilde{c}\Phi(\tilde{c})]_c^k+\int_{a}^{b} \left(\int_{c}^{k}\Phi(\tilde{c}) \mathrm d\tilde{c}\right) \omega(\tilde{p})\mathrm d (\tilde{p})$$
$$=[\Phi(k)]\bar{p}- [\tilde{c}\Phi(\tilde{c})]_c^k+\int_{a}^{b} \left(\Psi(k)\right)\omega(\tilde{p}) \mathrm d (\tilde{p}) \text{ where } \Psi'(\tilde{c})=\Phi(\tilde{c})$$
$$=[\Phi(k)]\bar{p}-[\tilde{c}\Phi(\tilde{c})]_c^k+\left(\Psi(k)\right)$$
$$\\=[\Phi(k)]\bar{p}-[k\Phi(k)]+\left(\Psi(k)\right)\\$$