You are making this way too complicated.
From inspection you can see that during the first half of the period shown, you have a steady voltage of 500 mV. The power is then just the average of the current times this voltage, which is 250 mW. From inspection again you can see that the second half of the period is the same as the first half with the signs of the voltage and current flipped. This obviously yields the same power again, 250 mW.
The instantaneous power is a triangle wave with peaks at 0 and 500 mW, and average of 250 mW (unless I'm misunderstanding what that diagram is showing).
Added:
I forgot to mention about calculating reative power.
One way to get that is to derive the power factor. The power factor is usually described as the cosine of the phase angle between the current and voltage assuming both are sines. However, it also has a more general definition that is more appropriate in this case. You can think of the power factor as the ratio of true power to the product of RMS current and voltage.
In this case, the RMS voltage is obvious, which is 500 mV. From inspection you can see that the current is symmetric and repeating, so you only have to solve for the RMS current of a ramp from 1 to 0. From symmetry we can see that this must be the same as a ramp from 0 to 1, which will make the equation a little easier.
In other words, find the RMS current of I(t) = t from 0 to 1. To do that, first square the function, which is then t^2. The average of that from 0 to 1 is 1/3, and then the square root of that is 0.577. So the RMS voltage is 500 mV, the RMS current is 577 mA, and the product of the two is 289 mW. From above the real power is only 250 mW, so the power factor is 250mW/289mW = 0.866. The reactive power is
sqrt(289mW^2 - 250mW^2) = 144 mW
Again, there is no need to make this complicated.
If it's a duty cycle of 4%, then the average load is 4% of 1.95mA, or 78 uA. The battery should last 2610 / 0.078 hours, or 33461 hours. Over 3 years.
Battery capacity changes with temperature, time, and discharge rate (and the 2610 mAh rating isn't too precise either, in my experience.) These calculations are probably within ±20% of the real answer.
Best Answer
This is easiest done in two parts, for the first and second halves of the repeating waveform. This time the average power will obviously be negative since the power magnitude in the second half clearly is larger than in the first half, and the first half is obviously positive and the second negative. This means the load is supplying net power.
The total power magnitude is the RMS voltage times the RMS current. The RMS voltage is obviously 1 V. The current is a ramp from 0 to 1, or I(t) = t. That squared is t^2, the average of that is 1/3, and the square root of that is 0.577. The power magnitude is therefore 1 V * 577 mA = 577 mW.
The real power is the integral of the instantaneous voltage and current over one repeating cycle. From inspection, this is 250 mW in the first half and -750 mW in the second half, for a total of -500 mW.
The power factor is -500 mW / 577 mW = -0.866. Often the absolute value of the power factor is quoted with a statement as to whether the reactive load is inductive or capacitive. You can take the arc cosine of the power factor to get the phase angle of -30 deg in this case, but that has little meaning when the voltage and current are so non-sinusoidal.
The total complex power is the vector sum of the real and reactive power. Since we know the real power and the power magnitude, we can calculate the reactive power, which is -289 mW. The sign only indicates whether it is capacitive or inductive (whether it is a +90 deg or -90 deg).