# Fundamental Theorem of Calculus

## Homework Statement

All this information is in the attached file.

## Homework Equations

All this information is in the attached file.

## The Attempt at a Solution

What I tried to do was take the anti-derivative of the first equation and plug in the number 5. I'm not sure if that was the way to go so I then just plugged in 5 off the get-go and got 13. Not sure what the question is asking and I would appreciate some guidance.

Thanks

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Mark44
Mentor

## Homework Statement

All this information is in the attached file.

## Homework Equations

All this information is in the attached file.

## The Attempt at a Solution

What I tried to do was take the anti-derivative of the first equation and plug in the number 5. I'm not sure if that was the way to go so I then just plugged in 5 off the get-go and got 13. Not sure what the question is asking and I would appreciate some guidance.

Thanks
If you have concluded that F(5) = 13, then no, that is incorrect. For that part you should NOT attempt to get the antiderivative.

So the anti-derivative is only applicable to parts b and c?

Mark44
Mentor
You should not attempt to get the antiderivative in any of the three parts. For part b, you'll need to use the Fundamental Theorem of Calculus. Note that there are two parts of this theorem.

The book says to use the second part of the theorem but I'm not seeing how it would work in this question. Do I plug in the x in place of t?

Mark44
Mentor
You're talking about part a, right? What definite integral is represented by F(5)?

√5^2 + 144?

Mark44
Mentor
√5^2 + 144?
No.
Show me what the definite integral looks like. If you quote this post you should be able to reuse this integral.

$$F(5) = \int_?^? \sqrt{t^2 + 144}~dt$$

No.
Show me what the definite integral looks like. If you quote this post you should be able to reuse this integral.

$$F(5) = \int_5^5 \sqrt{t^2 + 144}~dt$$

Is it that?

Mark44
Mentor
Yes. Now, what is the value of that integral? That is what F(5) equals. Don't overthink this.

SammyS
Staff Emeritus
Homework Helper
Gold Member
ralfsk8,

In general, what does the definite integral,
$\displaystyle \int_{a}^{b}\, f(x)\,dx$​
represent?

Mark 44 - I would say either 13 or 0, but as you mentioned earlier, 13 isn't the answer.

SammyS - The area under the curve bounded by the x-axis of a graph and 2 vertical lines (a and b)

Mark44
Mentor
What are the vertical lines for this integral?

The vertical lines for this integral are 5 and x

Mark44
Mentor
That's for F(x). You're trying to determine F(5). So what are the vertical lines now?

STEMucator
Homework Helper
An easy way to remember the fundamental theorem is to consider :

Let :

$F(x) = \displaystyle \int_{b(x)}^{a(x)}\, f(t)\,dt$

Then :

$\frac{d}{dx}F(x) = \frac{d}{dx} \displaystyle \int_{b(x)}^{a(x)}\, f(t)\,dt = f(a(x))a'(x) - f(b(x))b'(x)$

It's simply a plug and play formula.

Mark44 - The vertical lines are now 5 and 5. So we're not trying to calculate the area of anything if it's the same line?

Zondrina - Ah yes, that does seem familiar to me

Mark44 - I would assume 5 to 5 which would make it the same line right?

Zondrina - Ah yes, that does seem familiar

STEMucator
Homework Helper
Mark44 - I would assume 5 to 5 which would make it the same line right?

Zondrina - Ah yes, that does seem familiar

Also think about this F(5) thing, it's MUCH easier than it looks. Imagine you're integrating from one line a to another line b. What if those two lines are the same? Then there really is no area between them right?

A simple example is to consider :

$\int_{z}^{z}xdx$

Where z is ANY real number or function or anything really, but to keep it simple let z = 1,2,3...... and notice no matter what you're going to get the same answer. Then come back to your F(5) question.

Mark44
Mentor
Mark44 - I would assume 5 to 5 which would make it the same line right?
Right. So F(5) = ?

Okay so I've narrowed it down to either 0 or 5...

Mark44
Mentor
Okay so I've narrowed it down to either 0 or 5...
What's your reasoning for each of these?

Well just like Zondrina said, if both of the vertical lines have the same value, then there shouldn't be area, deeming it 0.

As for the 5... I actually have no clue

Mark44
Mentor
Well just like Zondrina said, if both of the vertical lines have the same value, then there shouldn't be area, deeming it 0.

As for the 5... I actually have no clue

Mark44
Mentor
Right.

Now, what about parts b and c?

Do I take the anti-derivative and plug in 5?

STEMucator
Homework Helper
Do I take the anti-derivative and plug in 5?

No you use that plug and play i posted on the first page. Memorize it as it is your best friend to solve questions like these within a minute.

Mark44
Mentor
ralfsk8 said:
Do I take the anti-derivative and plug in 5?

No. Here's what I said back in post #4.
You should not attempt to get the antiderivative in any of the three parts. For part b, you'll need to use the Fundamental Theorem of Calculus. Note that there are two parts of this theorem.

Mark44
Mentor
No you use that plug and play i posted on the first page. Memorize it as it is your best friend to solve questions like these within a minute.
IMHO, memorizing that formula is NOT a good idea, especially if that memorization comes at the expense of understanding.