I am unable to get my resultant integral to sum to unity

Questions
1

Friends,

I have two piecewise functions: fx[t] and fy[t]. Both functions are intended to be density functions. The area under both functions sum to unity.

I need to convolve the two functions. The resultant function spans the interval [0,1/2+Sqrt[3]/2]. I have done the convolution via both Laplace transform and tradition integration. Both solutions look ok UNTIL i check the area under the resultant function. I does NOT sum to unity. Instead it sums to about 0.9994 or something like that.

Does anyone know what I might be doing wrong? I have read online, but I just cannot fix the problem. Perhaps my Assumptions are not right???

My work is attached.

Open in Wolfram Cloud ↗
2

fx[t] is normalized over [0,1], but the but fy[t] is normalized over [ 0, Sqrt[3]/2]

In[10]:= Integrate[fx[t], {t, 0, 1}]

Out[10]= 1

In[11]:= Integrate[fy[t], {t, 0, Sqrt[3]/2}]

Out[11]= 1

In[13]:= Integrate[fx[t], {t, 0, Sqrt[3]/2}] // N

Out[13]= 0.99957
3

Your upper integration bound needs to be 1 + Sqrt[3]/2 since that is the largest value that X + Y could take

Integrate[dL, {t, 0, 1 + Sqrt[3]/2}]
(*1*)

Integrate[dRect[t], {t, 0, 1 + Sqrt[3]/2}]
(*1*)