Theory question Differentiation / Integration

[orochimaru]

Homework Statement

a) Write down the definition of the definite integral in terms of a limiting procedure of elementary areas

b) Write down the definition of the indefinate integral

c) Show that the derivative of an indefinite integral of f(x) is f(x)

The Attempt at a Solution

I'm not really sure what they mean in part a. Is it something like if there is a function y(x) the area under the curve between limits x=b and x=a where
b > a. If it is divided into strips of Area $$\ y \delta x = \delta A$$ then $$\ y = limit \delta x$$ goes to 0 $$\frac{\delta x}{\delta a} = \frac{dA}{dx}$$
so A = integral from x=b to x=a ydx

An indefinite integral is a family of functions ie. F(x) + C whose derivatives are all f(x)

If the integral of f(x)dx is F(x) + C , then the derivative of F(x) + C is f(x) + 0
= f(x)

I'm really not sure about these, could someone help me out please.

 

[HallsofIvy]

The first two are asking for definitions. They are really just asking you to quote your textbook. (c) is part of the "fundamental theorem of calculus". Look it up in your textbook.

 

[orochimaru]

I think I'm ok with part a and b now but my textbook doesn't have the fundamental theorem of calculus in it. I have looked on the internet but all of the proofs for part c give the integral limits and an indefinite integral doesn't have limits.

for a my guess is something like

$$\lim_{\delta x\to0} \sum_{x=a}^{x=b} f(x) \delta x = <br /> \int_a^b f(x) \,dx$$

The bottom limit of the summation is supposed to be a but for some reason I can't edit it

for b I got this definition from the internet
the set of functions F(x) + C, where C is any real number, such that F(x) is the integral of f(x)

I would still appreciate help with part c though

 

[Gib Z]

The definition you have for b was good but is circular..

Try something like: The set of functions F(x)+ C, where C is any real number, such that the derivative of F(x) is f(x).

C follows straight from that definition.

 

[orochimaru]

I've been told by my teacher that we are supposed to use the fundamental theorem of calculus for part c

so for part b I define the indefinite integral as

$$\ F(x) = \int_a^x f(t)dt$$ = F(x) - F(a) = F(x) +C

and for part c

I take $$\int_x^{x+h} f(t) dt = F(x+h) - F(x) \approx f(x)h$$

so $$\frac {F(x+h)-F(x)}{h} \approx f(x)$$

then $$\lim{h\to0} \frac {F(x+h)-F(x)}{h} = F^{'}(x) = f(x)$$

 

[Gib Z]

I would have defined the indefinite integral as in my post #4. For C, I would have done this:

The indefinite integral is defined to be The set of functions F(x)+ C, where C is any real number, such that the derivative of F(x) is f(x).

So by definition, the derivative of F(x) is f(x).