Prove this Definite-Interval rule: int[a,b]f(x)dx = int[a,c]f(x)dx + int[c,b]f(x)dx

[dan34]

I have to prove that,

If \(\displaystyle f\, :\, [a,\, b]\, \rightarrow\, \mathbb{R}\) is continuous and c is an element of (a, b), then

. . . . .\(\displaystyle \displaystyle \int_a^b\, f(x)\, dx\, =\, \int_a^c\, f(x)\, dx\, +\, \int_c^b\, f(x)\, dx\)

Hint: Since f is supposed to be continuous, you may use the definition of the limit of the Riemann sum for regular partitions. <- this one

This is how I am thinking, and please tell me if this is the right thing to do or not:


Take the limit of Riemann for the left expression, since we know that C is in somewhere in the middle of A and B, we can divide the limit of sum such that the first lim SUM goes from a to c + lim SUM goes from c to b, and then from these 2 limits and I can write that Integral from a to c + Integral from c to b = Integral from a to b.

 

[stapel]

I have to prove that,

If \(\displaystyle f\, :\, [a,\, b]\, \rightarrow\, \mathbb{R}\) is continuous and c is an element of (a, b), then

. . . . .\(\displaystyle \displaystyle \int_a^b\, f(x)\, dx\, =\, \int_a^c\, f(x)\, dx\, +\, \int_c^b\, f(x)\, dx\)

Hint: Since f is supposed to be continuous, you may use the definition of the limit of the Riemann sum for regular partitions. <- this one

This is how I am thinking, and please tell me if this is the right thing to do or not:


Take the limit of Riemann for the left expression, since we know that C is in somewhere in the middle of A and B, we can divide the limit of sum such that the first lim SUM goes from a to c + lim SUM goes from c to b, and then from these 2 limits and I can write that Integral from a to c + Integral from c to b = Integral from a to b.

I think you're at least on the right track (regarding whatever it is that the instructor is expected from you). As near as I can tell, your method is pretty much what everybody uses, in some form or another (page 10).