Hello, I'm getting slightly confused by the following so was hoping someone may be able to clear my problem up.(adsbygoogle = window.adsbygoogle || []).push({});

For integrals, if b is the upper limit and a is the lower limit, I will write ∫[b,a].

From the Fundamental Theorem of calculus part 1 we can show that:

if

F(x) = ∫[x,a] f(t)dt

then

F'(x) = f(x)

where F'(x) is the derivative of F(x) with respect to x

I understand the proof so will not detail.

From this, we can deduce part 2, that:

if

G(x) is any anti-derivative of f(x)

then

∫[b,a] f(t)dt = G(b)-G(a) = G(x) (evaluated at b) - G(x) (evaluated at a)

I understand the proof here also.

However, what I don't understand is that part 2 is actually written:

∫[b,a] f(x)dx = G(b)-G(a)

This notation suggests that integrating with respect to the x, being the term after the integrand is acceptable. However, we actually obtain the proof by finding the anti-derivative w.r.t. x when x was the upper limit in the summation.

My question is - what is the step that seems to be missing seeing as in the second law we actually integrate with respect to what was previously denoted as t? I understand that x and t are two different variables but they are obviously closely related - is it a case that t is dependent on x? If we can integrate w.r.t. either x or t surely they are equal and vary on a 1 to 1 ratio?

Thanks for the help,

James

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# Fundamental Theorem of Calculus - Variables x and t

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