Question

Calculus 2 - Second theorem of calculus

Answer 1

This is solved using the `First Fundamental Theorem of Calculus`.
Err at least that's the way I remember learning it, and how wiki lists them.

Maybe your teacher taught you in the reverse order, no matter.
So we'll use the fact that:\[\Large\bf\sf \frac{d}{dx}\int\limits_a^xf(t)\;dt \quad=\quad f(x)\]

Answer 2

Ahh crap I forgot the equation tool isn't working right..
Is my math equation showing up as a big mess of code?

Answer 3

yea i can't see it

Answer 4

I guess I can post a picture of the steps.
It'll be way too difficult to explain in text.
Gimme a couple mins.

Answer 5

Thank you

Answer 6

Here is the rule we want to apply:

Answer 7

So take a look at that a moment, lemme know if anything doesn't make sense.
It's a weird process.

We're bypassing the step where we would `actually` find an anti-derivative because our next step is to find the derivative.

So we're integrating, then undoing the integration by differentiating.
So there is no need to do the intermediate step of actually finding the anti-derivative.

Answer 8

sorry, what do you mean?

Answer 9

Does the concept in that third picture make sense?

Answer 10

yes

Answer 11

So the idea is, we start with a function f(x).
Then we do 2 things to it,
~Integrate
~Differentiate

These 2 operations lead us back to where we started, with good ole f(x).

We don't need to actually FIND the anti-derivative and then take it's derivative.
We can follow the rule that tells us we'll end up with the thing we started with.

Answer 12

In picture 2, I took that big ugly natural long and called it a function f(t).

Answer 13

We "integrate" it, giving us F(t). (Whatever that may be).

We then plug in our boundaries and now that x is inside of our function we can differentiate.

Answer 14

So the rule in picture 3, I hope makes sense.

The only difference in our problem is,
The variable changes from t to x in the middle of the problem.
This is important since our derivative operator is with respect to x.

Answer 15

Is this not making any sense? :o tricky stuff?

Answer 16

sorta, it is tricky lol

Answer 17

thank you

Answer 18

hmm darn :c

Answer 19

Sorry I'm not quite sure how to get this concept across.
It's always a little tricky for students.
Here's another shot at it :\
Maybe try to make some sense of this picture.