We have the Random Variable X, which is $\mathrm{\Gamma}(p,\lambda )$ distributed with the density:

$${f}_{p,\lambda}(x)=\frac{{\lambda}^{p}}{\mathrm{\Gamma}(p)}\cdot {x}^{p-1}\cdot {e}^{-\lambda x}$$

with p=10 and ${H}_{0}:\lambda =2$ or ${H}_{1}:\lambda =4$ and $\alpha =0.001$

I want to apply the Lemma of Neyman Pearson which states:

Be c>0 fixed and chosen in the way that $A(c)=\{x\in B:\frac{{f}_{0}(x)}{{f}_{1}(x)}\ge c\}$ such that ${\mathbb{P}}_{{H}_{0}}(X\in A(c))=\alpha $

Then the test with the region A(c) among all tests with significance level $\alpha $ is the most powerful.

I am now trying to calculate A(c), but got stuck. I have:

$${\int}_{A(c)}{f}_{0}(x)dx={\int}_{A(c)}\frac{{\lambda}^{p}}{\mathrm{\Gamma}(p)}\cdot {x}^{p-1}\cdot {e}^{-\lambda x}dx=\alpha .$$

But I don't know how to get A(c) from this integral...

$\frac{{f}_{0}(x)}{{f}_{1}(x)}=\frac{1}{1024}\cdot {e}^{2x}$

$${f}_{p,\lambda}(x)=\frac{{\lambda}^{p}}{\mathrm{\Gamma}(p)}\cdot {x}^{p-1}\cdot {e}^{-\lambda x}$$

with p=10 and ${H}_{0}:\lambda =2$ or ${H}_{1}:\lambda =4$ and $\alpha =0.001$

I want to apply the Lemma of Neyman Pearson which states:

Be c>0 fixed and chosen in the way that $A(c)=\{x\in B:\frac{{f}_{0}(x)}{{f}_{1}(x)}\ge c\}$ such that ${\mathbb{P}}_{{H}_{0}}(X\in A(c))=\alpha $

Then the test with the region A(c) among all tests with significance level $\alpha $ is the most powerful.

I am now trying to calculate A(c), but got stuck. I have:

$${\int}_{A(c)}{f}_{0}(x)dx={\int}_{A(c)}\frac{{\lambda}^{p}}{\mathrm{\Gamma}(p)}\cdot {x}^{p-1}\cdot {e}^{-\lambda x}dx=\alpha .$$

But I don't know how to get A(c) from this integral...

$\frac{{f}_{0}(x)}{{f}_{1}(x)}=\frac{1}{1024}\cdot {e}^{2x}$