Single vs. Double Integration for Area Bounded by y=x and y=x^2-2x

• kasse
In summary, the area of the region bounded by the graphs y=x and y=x^2-2x can be computed by finding the integral of (x-(x^2-2x)) dx from x=0 to x=3. Double integration is not necessary for this calculation. The total area is given by Int |f(x)-g(x)|dx and "ordinary" integration will suffice. This method can also be simplified to a single integral form.

kasse

If I want to compute the area of the region bounded by the graphs y=x and y=x^2-2x, can I simply compute the integral of (x-(x^2-2x)) dx from x=0 to x=3, or do I have to use double integration?

In any case, why?

For two curves, f(x) and g(x) the area of the element between x and x+dx is simply |f(x)-g(x)|dx. The magnitude sign is there only if you don't want the area to go negative when the curves cross.

So the total area = Int |f(x)-g(x)|dx

"Ordinary" integration will do.

I'm a bit confused by the question. Since you know about "double integration", you must have been studying calculus for some time- and should have learned to find the "area between two curves" quite a while ago!

Here, y= x is above y= x2- 2x for x between 0 and 3 so the area is given by
$$\int_0^3 (x- (x^2- 2x))dx= \int_0^3 (3x- x^2)dx$$

Of course, you could use "double integration" with '1' as integrand:
$$\int_{x=0}^3 \int_{y= x^2- 2x}^x dy dx$$
but after the first integration, it reduces to the single integral form.

1. What is the difference between single and double integration for finding the area bounded by y=x and y=x^2-2x?

Single integration calculates the area under a curve by finding the definite integral of the function. Double integration, on the other hand, calculates the volume under a surface by finding the double integral of the function.

2. Which method should I use to find the area bounded by y=x and y=x^2-2x?

This depends on what you are trying to find. If you are looking for the area under the curve, use single integration. If you are looking for the volume under the surface, use double integration.

3. Can I use both single and double integration for the same problem?

Yes, you can use both methods to solve the same problem. However, the results may differ depending on what you are trying to find.

4. How do I set up the integrals for single and double integration in this scenario?

For single integration, you will need to find the definite integral of the function from the lower bound to the upper bound. For double integration, you will need to set up a double integral with the function as the integrand and the bounds for the x and y variables.

5. Are there any advantages of using double integration over single integration for finding the area bounded by y=x and y=x^2-2x?

Double integration allows for more flexibility in finding the area or volume of more complex shapes, such as regions bounded by two curves. It also allows for more accurate calculations as it takes into account the entire surface rather than just the curve.