The integral of a sum?

Question

I was thinking if i could calculate the integral of a sum ... and i started by a simple function ... f(x)=x; and to make it a little harder i tried to solve for a; And here is the integral. ... integral (from 0 to a) of the sum (from x=1 to a) of x. All is equal to 6. Here are the steps i passed through to conclude that "a" is equal to 2. So i want you to let me know if i did it correctly or no ?... and if it is correct ... then what is it's meaning ? (The area of a SUM ?? ) Thank you for your help.
PS : "a" is an integer.

Answer 1

Your work is almost correct. That is, upto the integration and summation.

However, you must remember that a cubic equation has 3 roots, not just one. So the last step needs to be modified and the equation will be as follows:

$$a^3+a^2-12=0$$ $$(a-2)(a^2+3a+6)=0$$

So the roots of the quadratic equation are: $$a=\frac{-3\pm\sqrt{9-4\times1\times 6}}{2\cdot 1}$$ $$a=\frac{-3\pm i\sqrt{15}}{2}$$

So we get 2 complex conjugate roots. But since it is given that $a$ is an integer, so these two complex roots are not possible. $a = 2$ is the only answer.

Hope this helps you.