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So I was just working through Courant's calculus and am a bit confused as to where a few variables are pulled out of.
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 εi the right-hand end point of each interval we find the integral as the limit as n -> ∞ of the sum
Fn = (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)h2
= nah + (1/2)n(n+1)h2
(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.)
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 εi the right-hand end point of each interval we find the integral as the limit as n -> ∞ of the sum
Fn = (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)h2
= nah + (1/2)n(n+1)h2
(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.)