Help me Understand This Area

[nasi112]

I used to think that integration always represented the area under a curve. Recently, I encountered some problems involving area calculations, and only after a few attempts did I realize that parts of the graph were below the x-axis, causing the positive and negative areas to cancel each other out.

Since I am not very experienced at visualizing the graph of f(x), whenever I am asked to find the area under a curve, my instinct is to simply compute the integration.

That is how I approached these problems every time, assuming that the integration would directly give the area. Area = [math]\int_{a}^{b} f(x)[/math]
I solved this yesterday

[math]\int_{0}^{3} x(x-2)(x+2) dx = 2.25[/math]
The area happened to be 10.25 squared inches. I do not understand when and when not to include the negative area?

 

[Dr.Peterson]

I do not understand when and when not to include the negative area?

I may be missing your question; but when you're asked for a definite integral, you just do that; when you're asked for the area between a curve and the x-axis, you find where it's positive or negative and integrate those (or, equivalently, integrate the absolute value).

If you're asked to find the area under a curve, and it is not always positive, then there's a little ambiguity in the problem, and you ask.

 

[fresh_42]

You have to integrate from the left boundary to the next zero, to the next zero, etc., and finally from the right-most zero to the right boundary, and add all absolute values of those results in order to get an area.

The reason is that areas are always positive. Integrals, on the other side are oriented volumes. They can be positive or negative depending on whether you go clockwise or counterclockwise through the area. It becomes obvious in the formula
[math] \int_a^a f(x)\,dx=0= \int_a^b f(x)\,dx-\int_a^b f(x)\,dx=\int_a^b f(x)\,dx+\int_b^a f(x)\,dx\,.[/math]
Hence, getting an area means avoiding that the partial sections cancel each other. Look at the integral [imath] \displaystyle{\int_{-1}^1 x^3\,dx} [/imath] to visualize the situation.