Why Qayanat calculus and Approximate integral are correct ?

We have numbers. Fingers.

Then we use ruler or scale.

We measure 7cm.

Our eyes see 1+2+3+4+5+6+7. Though only 7 cm. And We come out with an accurate formula of (n(n+1))/2.

Next step.

We see inches. We don't get confused.  We multiply. By 2.54.

Multiply.

And (n(n+1))/2 is multiplied. We come out with accurate formula of ((2n)!) =2(1)2(1+2+3)2(1+2+3+4+5) . . . .2(1+2+3+4+5+....(2n-1))

We come out with accurate formula in both cases. Possibly proudness. An internal thing.

Qayanat calculus is above psychology. 

Approximate integral the reverse.