Integrals Explained: What Does t and dt Mean?

[xeon123]

I've this example of an integral.

\int^{x}_{a} f(t)dt

What t and dt means? Is there a relation between t and dt?

 

[HallsofIvy]

Since you are talking about an integral I assume you have some idea of what an integral is, either taking a course in Calculus or are reading a book on Calculus.

Every textbook I have seen introduces the integral in terms of the Riemann sum in which you divide an area under the graph of y=f(x) into very thin rectangles- each having base \Delta x and height f(x_i^*) for x_i^* being some x value within the base of the ith rectangle. The approximate area would be the sum of the areas of those rectangles: \sum f(x_i^*)\Delta x. One can show that the exact area is the limit of that as the base of those rectangles goes to 0. Then in the form \int f(x)dx or \int f(t)dt x and t are the independent variables in the functions f(x) and f(t), the limits of f(x_i^*) and f(t_i^*), and dx and dt are "infinitesmal" sections of the axis.

 

[xeon123]

So what's the purpose of having dx in the expression? If I'm correct, dx means the derivative of x. And, in each point of x in the interval \Delta x, x will be a fixed value (is it a constant?), and the derivative of a constant is always 0.

 

[micromass]

The "dx" is just a notation. It doesn't mean anything. It's just used to denote which variable we integrate.

For example

\int_0^1 (t+x)dx

means that we integrate with respect to x. While

\int_0^1 (t+x)dt

means that we integrate with respect to t.

 

[xeon123]

As far as I understand, in the first case we will only integrate x, and in the second case, t.

So, being f a function, f=(t+x)

In the first case, what happens to x? Can you integrate this example, please?

 

[micromass]

With the integral

\int (t+x)dx

we have that t is a constant and x is the integration variable, so

\int (t+x)dx = tx+\frac{x^2}{2}+C

While in

\int (t+x)dt

we treat x as a constant. Thus

\int (t+x)dx = \frac{t^2}{2}+tx+C

 

[xeon123]

Thanks, now I got it.

 

[hunt_mat]

I wrote some notes on integration, they should be available on the Math & Science Learning Materials section under notes on integration.