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## Homework Statement

Integration of f(x) = x

We can see that a trapezoid is formed, so the relevant equation:

1/2(b-a)(b+a) is the value of this integral.

To confirm that our limiting process leads analytically to the same result, we subdivide the interval from a to b into n equal parts by means of the points of division

a + h, a + 2h, . . ., a + (n-1)h, where h = (b-a)/n.

(I still understand at this point as this is simply diving into n pieces)

Taking for ε

_{i}the right-hand end point of each interval we find the integral as the limit as n -> ∞ of the sum

F

_{n}= (a+h)h + (a+2h)h + . . . + (a + nh)h

(At this point I am not sure where the h outside the brackets has come from and what it represents, I thought h was the distance between segments?)

= nah + (1+2+3+ . . . + n)h

^{2}

= nah + (1/2)n(n+1)h

^{2}

(And at this point I am basically completely lost, I know the arithmetic series formula applies, but do not understand how to get to this point.)