Which function should be used when dealing with two curves in an integral?

In summary, when finding the area between two curves, the formula calls for integrating f(x) - g(x). When the formula calls for f^2(x), it depends on the specific problem and method being used. For finding the volume of revolution, one method involves integrating πf(x)^2dx and for the volume of a washer, the formula is π(f(x)^2 - g(x)^2)dx. It is important to understand the reasoning behind any formula rather than simply memorizing it.
  • #1
tandoorichicken
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When you are trying to find an integral and you are dealing with two curves, and the formula calls for [itex] f(x) [/itex], I know that you do [itex] f(x) - g(x) [/tex] and put in that value for the f(x) in the original formula.

When the formula calls for [itex] f^2 (x) [/itex], do you do [itex] (f^2 (x) - g^2 (x)) [/itex] or [itex] (f(x)-g(x))^2 [/itex]
?
 
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  • #2
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 [itex] f^2 (x) [/itex]
, do you do [itex] (f^2 (x) - g^2 (x)) [/itex]
or [itex] (f(x)-g(x))^2 [/itex]?"

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 πf(x)2dx because you are thinking of f(x) as the radius of a circle so that πf(x)2 is the area of the circle and πf(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: πf(x)2- πg(x)2. Then the volume is the integral of π(f(x)2- g(x)2)dx.
But you never just "do" something without understanding why you do it.
 
  • #3


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.
 

1. What is the significance of two curves in an integral?

The two curves in an integral represent the boundaries of the region that is being integrated. The area between the two curves is calculated by the integral, and can have various interpretations depending on the context of the problem.

2. How do you determine the limits of integration when there are two curves?

The limits of integration are determined by finding the points of intersection between the two curves. These points will serve as the boundaries for the region being integrated.

3. Can the two curves in an integral be any shape?

Yes, the two curves can be any shape as long as they intersect and form a closed region. The shape of the curves will affect the difficulty of the integration, but the principles remain the same.

4. How do you know which curve should be the upper and lower boundary?

This depends on the orientation of the region and the direction of integration. Generally, the upper curve will have a higher y-value for any given x-value, and the lower curve will have a lower y-value. However, it is important to check the direction of integration to ensure the correct orientation of the boundaries.

5. Are there any special techniques for integrating with two curves?

There are various techniques and strategies that can be used to integrate with two curves, such as using symmetry, breaking the region into smaller, simpler shapes, or using trigonometric identities. It is important to choose the method that best suits the problem at hand.

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