Should you square the difference or difference of squares in disk method integrals?

[tandoorichicken]

When you are trying to find an integral and you are dealing with two curves, and the formula calls for $f(x)$, I know that you do [itex]f(x) - g(x) [/tex] and put in that value for the f(x) in the original formula.<br /> <br /> When the formula calls for $f^2 (x)$, do you do $(f^2 (x) - g^2 (x))$ or $(f(x)-g(x))^2$<br /> ?[/itex]

 

[HallsofIvy]

This is what comes of memorizing formulas without understanding.

"When you are trying to find an integral and you are dealing with two curves". You understand, don't you, that an integral does not necessarily have anything to do with "two curves" I think you are talking about finding the area between two curves. In that case you integrate f(x)- g(x) because you are using that distance as the height of the "rectangles" in the Riemann sum.

"When the formula calls for $f^2 (x)$
, do you do $(f^2 (x) - g^2 (x))$
or $(f(x)-g(x))^2$?"

Well, I don't know because I have no idea what formula you are talking about or what kind of problem you are doing. I do recall that one method of finding a "volume of revolution" involves integrating &pi;f(x)²dx because you are thinking of f(x) as the radius of a circle so that &pi;f(x)² is the area of the circle and &pi;f(x)²dx is the volume of the flat disk.
If the axis of rotation is outside the figure, then you can think of it as one circle inside another (a "washer"). You could find the area of the washer by calculating the area of the outer circle and then subtracting the area of the inner circle: &pi;f(x)²- &pi;g(x)². Then the volume is the integral of &pi;(f(x)²- g(x)²)dx.
But you never just "do" something without understanding why you do it.

 

[M98Ranger]

When dealing with two curves in an integral, it is important to understand the difference between f(x) and f^2(x). In the first scenario, where the formula calls for f(x), you would indeed use the difference between the two curves, which is (f(x) - g(x)). This is because the integral is essentially finding the area between the two curves, and the difference between them represents the height of each rectangle used to approximate the area.

In the second scenario, where the formula calls for f^2(x), you would use (f(x))^2 - (g(x))^2. This is because the integral is now finding the area between the curves squared, so you need to square each individual function before taking the difference between them. Using (f(x)-g(x))^2 would not accurately represent the area between the curves squared.

It is important to carefully consider the formula and the concept being represented in order to correctly manipulate the functions within the integral. Remember to always follow the rules of integration and carefully consider the specific scenario at hand.