Fundamental theorem problem

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February 27th 2013, 06:53 AM
Steelers72

Fundamental theorem problem

http://www.webassign.net/cgi-perl/sy...E2%2B5%29%20du

Ok, so you split up into two intervals I did from 2x to 0 and 0 to 4x.

Then you reverse the 2x to 0 and the integral becomes negative.

Then, replace 2x and 4x with u and chain rule

answer: (-2(x)²-5/(2x)²+5)*2 + (4(x)²-5/(4x)²+5)*4

Is my work correct? I was marked wrong and it said we didnt have to simplify.
February 27th 2013, 07:00 AM
Plato

Re: Fundamental theorem problem

Quote:

Originally Posted by Steelers72

http://www.webassign.net/cgi-perl/sy...E2%2B5%29%20du

What the heck are you doing? What are the instructions?
February 27th 2013, 07:04 AM
HallsofIvy

Re: Fundamental theorem problem

Quote:

Originally Posted by Steelers72

http://www.webassign.net/cgi-perl/sy...E2%2B5%29%20du

Ok, so you split up into two intervals I did from 2x to 0 and 0 to 4x.

Then you reverse the 2x to 0 and the integral becomes negative.

Then, replace 2x and 4x with u and chain rule

answer: (-2(x)²-5/(2x)²+5)*2 + (4(x)²-5/(4x)²+5)*4

Did you intend this as an integral?

Quote:

Is my work correct? I was marked wrong and it said we didnt have to simplify.
February 27th 2013, 08:32 AM
Steelers72

Re: Fundamental theorem problem

Fundamental theorem, it says to find the derivative. So what I did was split up into two integrals because we have an x for both b and a of the integral. I split into intervals 2x to 0 and 0 to 4x. Since 2x is still on the bottom of integral, you move it by law of integrals to the top and it becomes a negative integral from 0 to 2x. Then I plugged in 2x and 4x in for each u value and did chain rule of 2x and 4x and got the answer from above. Sorry I don't know how to make integrals on the computer
February 27th 2013, 08:45 AM
Plato

Re: Fundamental theorem problem

Quote:

Originally Posted by Steelers72

Fundamental theorem, it says to find the derivative. So what I did was split up into two integrals because we have an x for both b and a of the integral. I split into intervals 2x to 0 and 0 to 4x. Since 2x is still on the bottom of integral, you move it by law of integrals to the top and it becomes a negative integral from 0 to 2x. Then I plugged in 2x and 4x in for each u value and did chain rule of 2x and 4x and got the answer from above. Sorry I don't know how to make integrals on the computer

There is absolutely no reason to 'split' anything.

Suppose that each of $g~\&~h$ is a differentiable function then if
$F(x) = \int_{h(x)}^{g(x)} {\Phi (t)dt}$ then $F'(x) = g'(x)\Phi (g(x)) - h'(x)\Phi (h(x))$ .
February 27th 2013, 08:56 AM
Steelers72

Re: Fundamental theorem problem

My professor told us to split that's why I did it because he said when you have an x for both the top and bottom intervals you should split into two separate integrals. I guess you are saying some sort of rule that explains this without splitting it up?
February 27th 2013, 09:02 AM
Plato

Re: Fundamental theorem problem

Quote:

Originally Posted by Steelers72

My professor told us to split that's why I did it because he said when you have an x for both the top and bottom intervals you should split into two separate integrals. I guess you are saying some sort of rule that explains this without splitting it up?

It is just a simple application of the chain rule.

Now it is true that $\int_{h(x)}^{g(x)} {\Phi (t)dt} = \int_0^{g(x)} {\Phi (t)dt} - \int_0^{h(x)} {\Phi (t)dt}$ .

But why waste time?
February 27th 2013, 09:13 AM
Steelers72

Re: Fundamental theorem problem

You are right but I think he didn't want to confuse us but he'll probably do it next class haha