# Definite integral of an absolute value function

• PFuser1232
In summary, a definite integral of an absolute value function is a mathematical operation that calculates the area under the curve of an absolute value function between two given points. To solve it, the function must be broken into two separate functions and the properties of integration can be used to find the overall area. The main difference between a definite and indefinite integral is that the former has specific limits while the latter represents the overall area. Applications of definite integrals include calculating displacement, velocity, and acceleration, as well as finding the average value of a function. The fundamental theorem of calculus can be used to find the definite integral of an absolute value function.
PFuser1232
Can we integrate:
$$\int_a^b |x| dx$$
using an antiderivative of ##|x|##, namely ##\frac{1}{2} x |x|##, instead of splitting up the integration interval?
I know this is not particularly useful for integrals such as:
$$\int_{-5}^5 |t^3 - 8| dt$$
However, for absolute value functions with linear arguments, this method (if valid) would be much more efficient.

Yes, of course. If F(x) is an anti-derivative of f(x) then $\int_a^b f(x) dx= F(b)- F(a)$. That is true for f(x)= |x| and F(x)= (1/2)x|x|.

## 1. What is a definite integral of an absolute value function?

A definite integral of an absolute value function is a mathematical operation that calculates the area under the curve of an absolute value function between two given points, also known as the definite integral.

## 2. How do you solve a definite integral of an absolute value function?

To solve a definite integral of an absolute value function, you must first break the function into two separate functions based on the point at which the absolute value changes sign. Then, you can use the properties of integration to find the area under each function and add them together to get the overall definite integral.

## 3. What is the difference between a definite integral and an indefinite integral of an absolute value function?

A definite integral of an absolute value function calculates the area under the curve between two specific points, while an indefinite integral of an absolute value function is a function that represents the area under the curve without specific limits.

## 4. What are the applications of definite integrals of absolute value functions?

Definite integrals of absolute value functions have various applications in physics, engineering, and economics. They can be used to calculate displacement, velocity, and acceleration of objects, as well as finding the average value of a function over a given interval.

## 5. Can you use the fundamental theorem of calculus to find the definite integral of an absolute value function?

Yes, the fundamental theorem of calculus can be used to find the definite integral of an absolute value function. This theorem states that the definite integral of a function can be found by evaluating the antiderivative of the function at the upper and lower limits of integration and subtracting the results.

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