Does the dx in an integral signify multiplication?

In a sense, yes. In finding the area under a curve, which is what the integral is, you sum up the areas of rectangles. The integral symbol is the degenerate S from sum, f(x) gives you the height of the rectangle for a particular value of x, and dx is the width of the rectangle. As dx shrinks to 0, the number of rectangles increases to infinity and the error (between the sum of the areas of the rectangles and the actual area under the curve) shrinks to 0.

Didn't you do Riemann sums and limits in precalc?

The dx, means "with respect to x", it refers the the axis on which we are taking the integration.

Remember when you learned about Implicit Differentiation.

You learned that you could take the derivative of different axes, such as the y and the z.

The same applies when you integrate.

When you first learn how to integrate we only use 1 axis, the x-axis. However we don't have to integrate just on the x-axis.

Thus we need to know which axis to integrate on.

For that reason we have a dx. This is the same dx found when we took a derivative and had "d / dx" Again it meant with respect to the x axis.

If you are curious...

Later in Calculus III you will learn to use multiple axes in 3D.

You will see dy, and dz.

For example,

If we were doing dy, we would treat all other variables as constants. So x and z would not be integrated, only the y would be.

For more information on 3D integration refer to the Calculus concept known as the "triple integral"

I will give you some links to get you started...

http://en.wikipedia.org/wiki/Multiple_integral

Note: the way the triple integral works, we integrate from the "inside out" meaning that we first integrate the limits on the farthest right, with the variable on the farthest left.

These examples should help...

http://www.math.umn.edu/~nykamp/m2374/readings/tri...

I hope that helps clear things up....

You need to be careful here, because dx is a differential, which obeys different rules than real values. But in a way, yes, it does signify multiplication.

I say this because if/when you get into higher levels of mathematics, you will learn about Differential Equations and Partial Differentiation. If you take an equation like:

dy/dx = y

The only way to solve is the Separation of Variables Method, which requires you to treat dy and dx like ordinary variables.

You also have to use rules like this when talking about functions of more than one variable, and getting the Partial Derivative of them with respect to different variables. I won't confuse you with any more than that.

Officially no. but there is an informal way to think of it as multiplication.

For a positive function f, think of "f(x) dx" as the area of an infinitessimally thin rectangle with height f(x) and width equal to dx, where dx is "infinitessimaly small". then the integral sign indicates a summation of all these infinitessimal strips of area, and gives the total area under the curve.

Of course, what an integral really is is the limit of finite sums of rectangle areas

f(x) * delta x

which have positive width. As the number of rectangles n approaches infinity, the width delta x goes to 0, and the limit of these sums of "more and more terms, each of which are getting smaller and smaller" is the integral.

Yes...you can think of it in this way.

If you set up an integral to compute the area of a rectangle say....you could set up an integral with dxdy.....the infinitesimally small area, just like taking the area of a normal rectangle. It may not be a true multiplication, but it behaves as such.

Make sense?

No, dx translates to with respect to x.

Some times there are multiple variables in a function and the dx tells you which variable you are integrating for.

Example int(6x dx) is simple, but what if it was int(y*x dx)? you just use y as a constant. int(y*x dy) you use x as a constant. int(y*x) you don't know what to do.

no. the dx is just there to symbolize a derivative