# Area between curves, with an absolute value function

• tangibleLime
In summary, the problem involves finding the area between two curves using the formula A = \int_a^b f(x)-g(x) dx. The curves are determined by plotting the equations and finding their points of intersection. The antiderivatives of the functions are then found and used to set up the integral. The solution is incorrect, possibly due to the presence of an absolute value sign in one of the functions, which requires splitting up the case when x is positive and negative and adding up the areas for both cases.
tangibleLime

## Homework Statement

Then find the area S of the region. (between the curves)

## Homework Equations

$$A = \int_a^b f(x)-g(x) dx$$

## The Attempt at a Solution

First I plotted both equations, determined the top and the bottom functions and found where they intersected (calculator).

Then I set up the problem according the the area formula above...
$$A = \int_{-10}^{10} ((|9x|)-(x^2-10))dx$$

Then I found the antiderivatives for each function...
$$(\frac{9x}{\sqrt{x^2}})-(\frac{x^3}{3}-10x)$$

Then I subtracted the two bounds...
$$((\frac{9(10)}{\sqrt{(10)^2}})-(\frac{(10)^3}{3}-10(10))) - ((\frac{9(-10)}{\sqrt{(-10)^2}})-(\frac{(-10)^3}{3}-10(-10)))$$

This gave me the final answer of $$\frac{-1346}{3}$$, which was wrong. I assume my problem is the absolute value sign... because this method works fine with other problems of the same nature that don't have an absolute value as one of the functions.

Any help would be appreciated.

Last edited by a moderator:
|x| is x when x > 0 and -x when x < 0.

Split up the case when x > 0 and x < 0 and add up the areas for both cases.

## 1. What is the formula for finding the area between two curves with an absolute value function?

The formula for finding the area between two curves with an absolute value function is ∫|f(x)| - |g(x)| dx, where f(x) and g(x) are the two functions and dx represents the infinitesimal change in x.

## 2. How do you graph two curves with an absolute value function?

To graph two curves with an absolute value function, you first need to plot the individual graphs of |f(x)| and |g(x)|. Then, for any values of x where both functions have a positive value, the area between the two curves will be equal to the difference between the two functions. For any values of x where one function has a positive value and the other has a negative value, the area between the two curves will be equal to the sum of the two functions. Finally, for any values of x where both functions have a negative value, the area between the two curves will be equal to the difference between the two functions multiplied by -1.

## 3. Can the area between two curves with an absolute value function be negative?

Yes, in some cases the area between two curves with an absolute value function can be negative. This occurs when the two curves intersect and the value of one function is greater than the other at that point. In this case, the area between the two curves will be equal to the difference between the two functions, which can result in a negative value.

## 4. Is there a way to find the area between two curves with an absolute value function without graphing?

Yes, there is a way to find the area between two curves with an absolute value function without graphing. You can use the formula ∫|f(x)| - |g(x)| dx and evaluate it using calculus techniques such as integration by parts or substitution. This will give you the exact value of the area without having to graph the functions.

## 5. Are there any real-life applications of finding the area between curves with an absolute value function?

Yes, there are several real-life applications of finding the area between curves with an absolute value function. One example is in economics, where this concept can be used to calculate the total revenue and total cost of a business to determine its profit. It can also be used in physics to calculate the work done by a varying force over a given distance. Additionally, this concept is used in statistics to find the area under a probability distribution curve to calculate the probability of a certain event occurring.

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