What is the area between two intersecting parabolas?

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In summary, the problem involves finding the area of the region enclosed by two parabolas, which intersect at 0 and 1. The derivative of one of the parabolas is used to find the area, but there is a mistake in the calculation resulting in a negative value. The correct calculation involves subtracting the bottom function from the top function and integrating with respect to x.
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Find the area of the region enclosed by the parabolas $$y = x^2 $$and $$y = 2x - x^2.$$

So they intersect at 0 and 1.

The derivative of $$y = 2x - x^2.$$ is $$\d{y}{x} = 2 - 2x$$

When I plug in 1 I get 0, and when I plug in 0, I get 2, so I subtract 2 from 0 and the area is -2. But the area should be positive, what am I doing wrong?
 
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  • #2
The width of each area element is $dx$ and the height is the top function minus the bottom function, thus:

\(\displaystyle A=\int_0^1 \left(2x-x^2\right)-\left(x^2\right)\,dx=2\int_0^1 x-x^2\,dx\)

What do you get?
 

Related to What is the area between two intersecting parabolas?

1. What is the formula for finding the area between two parabolas?

The formula for finding the area between two parabolas is ∫(f(x) - g(x))dx, where f(x) and g(x) are the equations of the two parabolas.

2. How do I determine which parabola is on top?

To determine which parabola is on top, graph the equations and observe which one is higher for the given interval. Alternatively, you can set the two equations equal to each other and solve for the x-values where they intersect. The parabola with a larger y-value at the intersection point is on top.

3. Can the area between two parabolas be negative?

No, the area between two parabolas cannot be negative. The area is always a positive value as it represents the space between the two curves on the x-y plane.

4. Are there any special cases when finding the area between two parabolas?

Yes, there are two special cases when finding the area between two parabolas. The first is when the two parabolas do not intersect. In this case, the area between them is simply the sum of the areas under each individual parabola. The second case is when the two parabolas have the same equation but different intervals. In this case, the area between them is zero.

5. Can calculus be used to find the area between two parabolas?

Yes, calculus can be used to find the area between two parabolas. By integrating the difference between the two equations, we can find the area between them. This method is more efficient and accurate compared to using geometric formulas, especially when the parabolas have complex equations.

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