Hi;

This is what I get plugging into that formula:

This is the correct answer. However, for the benefit of future viewers, I demonstrate a quick way to tackle such problems (without the need for memorising formulae). Recall the following theorem:

Let f be locally Riemann integrable over I, and let F: I -> R be an indefinite integral of f. Then:

(i) F is continuous on I;

(ii) F is differentiable at each interior point c ∈ I at which f is continuous, and satisfies F'(c) = f(c);

(iii) If f is continuous on I, then F is a *primitive* of f.

This theorem allows us to perform the following manipulations on Anakin's problem.

We let

, and put. Then:So by the chain rule:

.bob bundy wrote:

hi Anakin,

In other words you can substitute into the integral process any alternative variable.

Not strictly true -- you'd need the 't' to act as the dummy variable.

]]>Good luck at class.

]]>Thanks Bobbym (and Bob) for the help, I'm very grateful.

]]>differentiate under the integral sign. So, I am going with that.]]>

Also, I meant plugging the intervals into sqrt(1 + t^3) by canceling the derivative and integral out would yield a different answer from the one I got.

]]>This is what I get plugging into that formula:

]]>Okay, thanks. I appreciate it!

]]>I am not getting that answer, let me look at it agian.

]]>By looking at the link that you last posted, I was able to get the following solution. Could you tell me if it seems right?

Also, it would seem that simply cancelling the integral and derivative would lead to a different solution - one that is lacking the 4x and sin(x) in front of the square roots.

Which is the right one then?

]]>A Leibniz integral is an differentiation under the integral sign and is something else. Otherwise Bob is right just cancel the integral and the derivative.

Please look here:

]]>Bobbym: I mean differentiating the definite integral, with respect to x. Or at least that's what I think the question appears to be asking for.

]]>Do you mean differentiation under the integral sign?

]]>In other words you can substitute into the integral process any alternative variable.

So I cannot see why you should just 'cancel out' the integration with the differentiation and just sub in the limits.

This worked OK with a simple function, but, I confess, I'm not 100% sure on this. As the day moves ahead, hopefully we'll get a second opinion on this.

Bob

]]>I've looked into the http://en.wikipedia.org/wiki/Leibniz_integral_rule but I'm not sure if that is the method I should be employing, as I've never come across it before. Or it is possible that I have but like I said, this is a review assignment and I haven't been in a math course for over 2 semesters so I may have forgotten it.

Any ideas on how to get started? I tried using substitution to express √(1+t^3) using only x but to no avail as the integral of √(1+t^3) itself looks far too long and complicated to be required for such a straightforward question (using a math engine).

]]>