# Finding X to minimise summed integral of two equations

I have two equations. In this example I'll make them linear, and later see if the solution generalises to different forms of equation:

$$y_1=-0.1x + 0.65$$ $$y_2=0.15x - 0.825$$

I'm interesting in small ranges of these graphs, say 0.5-2.5 of $y_1$ and the same range +7 of $y_2$. We assume the value of the equation outside of its range is 0.

I want to find a value for x such that the summed integration of $y_1$ from that point up to the end of its valid range, and the integration of $y_2$ from $(x+7)$ down to the start of its valid range, is minimal.

I hope that makes sense. I've sort of fried my brain getting the problem into these terms, and now can't see the solution for the trees.

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Integrating both equations gives:

$$\int y_1 dx = -0.05x^2 + 0.65x$$ $$\int y_2 dx = 0.075x^2 - 0.825x$$

The summed integrations for some value of x with respect to $y_1$ is then:

$$s=\int_x^2.5 y_1 dx + \int_7.5^{x+7} y_2 dx$$

The integration for the upper bound of $y_1$ and the lower bound of $y_2$ are constants. We'll call them $u_1$ and $l_2$. Expanding the above equation gives:

$$s=-0.025x^2+0.425x+u_1+l_2+2.1$$

If $u_1=1.3125$ and $l_2=-1.96875$, the final equation we need to minimise is:

$$s=-0.025x^2+0.425x+1.44375$$

The minimum value of $s$ for the range 0.5-2.5 is 1.65 at $x=0.5$.

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