# Area between two curves problem

• dumbQuestion
In summary, the area between two curves can be found by taking the integral of the "right function" from -1 to 2.5, the integral of the "left function" from 2.5 to 4, and the absolute value of the integral of the "left function" from -1 to -2.
dumbQuestion
Hello,
I have a question regarding finding the area between two curves. I will link to a well known example that seems to show up in every calculus textbook!

http://tutorial.math.lamar.edu/Classes/CalcI/AreaBetweenCurves.aspx

In particular on that page, I am referencing Example 6. And in particular, the page evaluates this integral with the functions in terms of x first, and then at the end of the example, with the function in terms of y. My issue is when the area is being found between the two functions in terms of y. I just don't understand how this is giving us the true area, as a lot of this area falls on the left side of the y axis, so won't the area be "negative" there? i mean even just look at that example and see the line x=y+1. From the region of y=-1 to y=-2, the x values are negative. So say we were looking at the integral of this line from y = -2 to y= 4. What we are really getting is the area between the line and the y-axis from y = -1 to y = 4 and then subtracting off the area between the line and the y-axis from y = -1 to y =-2. Do you see where I'm going with this? Why does just taking the integral of f(x)-g(x) give us the actual shaded area vs just the "net area"? If we want the actual shaded area, why don't we take the integral of the "right function" from -1 to 2.5, then the integral of the "right function" minus the "left function" from 2.5 to 4, then the absolute value of the integral of the "left function" from -1 to 2.5, and the absolute value of the integral of the "left function" minus the "right function" from -1 to -2?By the way I know I'm wrong I'm not at all trying to say I'm right, I just want to understand what I'm not understanding

Last edited:
From y=-2 to -1, integrating both functions with respect to y will give negative answers for both, but the integral of the line will be less negative (and hence, greater) than the integral of the parabola, so the difference is still a positive value and you should still subtract the parabola's net area from the line's net area.

oooooohI am looking at it right now, and I can see what I was missing. Thank you very much!

## 1. What is the "Area between two curves" problem?

The "Area between two curves" problem refers to finding the area enclosed by two curves on a graph. This is often done in calculus and is used to solve various real-world problems in mathematics, physics, and engineering.

## 2. How do you find the area between two curves?

To find the area between two curves, you first need to identify the points of intersection between the two curves. Then, you can use either the definite integral or the method of slicing to calculate the area between the curves.

## 3. What is the difference between using the definite integral and the method of slicing?

The definite integral involves finding the antiderivative of the function and plugging in the endpoints of the interval between the two curves. The method of slicing involves breaking the area into small rectangles and adding their individual areas together.

## 4. Can the area between two curves be negative?

Yes, the area between two curves can be negative. This can occur when the top curve is below the bottom curve for a certain interval, resulting in a negative area. In this case, the negative area represents the displacement below the x-axis.

## 5. How is the area between two curves used in real life?

The area between two curves has many real-world applications, such as finding the volume of irregularly shaped objects, calculating the work done by a variable force, and determining the balance of supply and demand in economics. It is also used in fields such as engineering, physics, and finance to solve various problems involving two intersecting curves.

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