Dead Weight Loss (Tax) Problem
Given demand $D(p)=A-ap$, and $A,a&gt;0$ and a fixed price $0&lt;p_1&lt;A/a$ by some company.
My solution so far
CS is
$CS=\int_{p}^{A/a}D(p)dp=\int_{p}^{A/a}(A-ap)dp=\frac{1}{2a}(A-ap)^2=\frac{(A-ap)^2}{2a}$
Problem
Find the CS now with a tax $p_1+t&gt;A/a$
My solution so far
Just let $p_2=p_1+t&gt;A/a$. Insert $p_2$ into the previous result since the integration would be the same and we get
$\frac{(A-a(p+t))^2}{2a}$
Main Problem
Find the DWL
My solution so far
I get the general formulas
$(p_2-p_1)p_1-\int_{p_1}^{p_2}A-ap\;dp$ and $1/2 \cdot (p_2-p_1)\cdot (Q_1-Q_2)$
And then the answer with the second formula that
$1/2*((p+t)-p)*((A-ap)-(A-a*(p+t)))=1/2\cdot t(a(p+t)-ap)$
Is this approach correct throughout the taxation influence on CS and the DWL?
I think you are making it unnecessarily hard for yourself. You can calculate the deadweight loss (DWL) simply as the difference between consumer surplus pre-tax and post tax, and also by adding tax revenue as it is not necessarily wasted if government uses it, so that would be:
$$DWL =  \frac{(A−ap)^2}{2a} -  \frac{(A−a(p+t))^2}{2a} +t(A-a(p+t))= A t - \frac{1}{2} \left( a t (2 p + t) \right) + t(A-a(p+t)) $$
In addition if you have further equations that explicitly model supply you should also add to the deadweight loss the difference between producer surplus pre-tax and post tax but if that is not specified it is common to assume that supply is just flat line (so there is no producer surplus).