Unerstanding an Integration question

[gingermom]

Homework Statement

for -1≤x≤1, F(x) =∫sqrt(1-t^2) from -1 to x ( sorry don't know how to put the limits on the sign

a. What does F(1) represent geometrically?
b. Evaluate F(1)
c. Find F'(x)

Homework Equations

The Attempt at a Solution

Since my teacher never seems to give simple questions I am wondering if I am missing something in what is being asked.
a. I know this is a semicircle with radius of 1
b. Evaluate - F(1) - I would think this is just plugging in for x=1 which would be ∏/2
c. It seems like F'(x) would just be the integrand so F' (x) = sqrt(1-t^2)

I feel like maybe I am missing something or am I trying to make this harder than it is?

 

[SammyS]

Homework Statement

for -1≤x≤1, F(x) =∫sqrt(1-t^2) from -1 to x ( sorry don't know how to put the limits on the sign

a. What does F(1) represent geometrically?
b. Evaluate F(1)
c. Find F'(x)

Homework Equations

The Attempt at a Solution

Since my teacher never seems to give simple questions I am wondering if I am missing something in what is being asked.
a. I know this is a semicircle with radius of 1
b. Evaluate - F(1) - I would think this is just plugging in for x=1 which would be ∏/2
c. It seems like F'(x) would just be the integrand so F' (x) = sqrt(1-t^2)

I feel like maybe I am missing something or am I trying to make this harder than it is?

a.
F(1) is not a semicircle in and of itself. F(1) is just some number. What does that number represent geometrically? Yes, it's related to that semi-circle.

b.
That's right.

c.
You said: F' (x) = sqrt(1-t^2). That's not right. There is a different independent variable on the left compared to the right.

 

[gingermom]

Oh, so F(1) would be the area of the semicircle - for C I will have to think on that - Would I find the antiderivative using substitution and then find the derivative of that?

Will go back and review taking the integral with variable in the limits - thanks

 

[haruspex]

Oh, so F(1) would be the area of the semicircle - for C I will have to think on that - Would I find the antiderivative using substitution and then find the derivative of that?

It's simpler than that - use the fundamental theorem of calculus.
For writing limits in forum posts, you could simply use sup and sub: ∫_x=0¹. But it looks much better with LaTeX: $\int_{x=0}^{1}$.
If anyone posts LaTeX you can see how they did it (and copy it) by right-clicking on the text and selecting Show Math As->TeX commands. It doesn't show the controls which bracket the LaTeX. There are, to my knowledge, four ways of doing those. You can use TEX and /TEX, each inside square brackets [], which will put the LaTeX on a line by itself, or use ITEX and /ITEX if you just want it to be part of a longer line. There's a shorthand form for each of these. The first can be done with just a double dollar sign at each end ("$$", no square brackets); the second with a double hash symbol ("##", # being called a "pound sign" in US).

 

[Calu]

You may either use a substitution to find F'(x) or use the fundamental theorem of calculus.

Finding a suitable substitution would be faster in an exam situation. Can you spot one?

(I was taught this using substitution 2 years before I was taught the fundamental theorem of calculus).

 

[gingermom]

so since the upper limit is x it would F '(x) =sqrt(1-x^2) * d/dx X which would be 1 so the answer would be F'(x) = sqrt(1-x^2)

Is that right?

 

[SammyS]

so since the upper limit is x it would F '(x) =sqrt(1-x^2) * d/dx X which would be 1 so the answer would be F'(x) = sqrt(1-x^2)

Is that right?

Part of what you wrote is wrong or mis-stated, part is right.

When you wrote " ... which is 1 ... ", to what does which refer?

 

[gingermom]

Okay, I think I was making this way harder than it needed to be - since the integral is from -1 to x and the upper limit is not something like x^2, by the Fundamental Rule of Calculus I should just be able to substitute the x for the t. If the upper limit been a limit that involved a function like x^2, then I would have had to use the chain rule. Is that correct?

 

[SammyS]

Okay, I think I was making this way harder than it needed to be - since the integral is from -1 to x and the upper limit is not something like x^2, by the Fundamental Rule of Calculus I should just be able to substitute the x for the t. If the upper limit been a limit that involved a function like x^2, then I would have had to use the chain rule. Is that correct?

That's pretty much it.

Your original answer said

F' (x) = sqrt(1-t^2)​

but it should have said

F' (x) = sqrt(1-x^2) .​

That's all I was getting at for part c .