Finding Area between 2 functions

In summary, the concept of finding area between 2 functions involves calculating the enclosed area between two given functions over a specific interval. The purpose of this is to determine the total space enclosed, which has many practical applications. The steps involved include identifying the functions and interval, finding the difference between them, integrating the difference function, and evaluating the definite integral. Common techniques used include Riemann Sums, Integration by Substitution, and Integration by Parts. Some common mistakes to avoid are forgetting to take the absolute value, not including the bounds of the interval, and using the wrong integration method.
  • #1
tmt1
234
0
Hi,

I have this problem to find the area between 2 curves:

$y = x^2$

and

$y = \frac{2}{x^2 +1}$

I found that the points of intersection are -1 and 1 and it is symmetrical.

I get
$2\int_{0}^{1} \ \frac{1}{x^2 + 1} - x^2 dx$which I am unable to solve. I have tried u-substitution but I end up getting mixed up.

Thanks
 
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  • #2
There's a standard integral involved:
$$\int \frac{dx}{x^2+1} = \arctan(x)$$

I think you can figure out the answer right now ;)
 

Related to Finding Area between 2 functions

What is the concept of finding area between 2 functions?

The concept of finding area between 2 functions is to calculate the area enclosed between two given functions on a particular interval. This can be done by finding the definite integral of the difference between the two functions over the given interval.

What is the purpose of finding the area between 2 functions?

The purpose of finding the area between 2 functions is to determine the total amount of space that is enclosed between the two functions on a given interval. This can be useful in many real-world applications such as calculating the total displacement of an object or determining the total profit of a business.

What are the steps involved in finding the area between 2 functions?

The steps involved in finding the area between 2 functions are as follows:

  1. Identify the two functions and the interval over which the area needs to be calculated.
  2. Find the difference between the two functions.
  3. Integrate the difference function over the given interval to find the definite integral.
  4. Evaluate the definite integral to find the area between the two functions.

What are some common techniques used to find the area between 2 functions?

Some common techniques used to find the area between 2 functions are:

  • Riemann Sums: This involves dividing the interval into smaller subintervals and approximating the area using rectangles.
  • Integration by Substitution: This involves substituting one variable with another to simplify the integral.
  • Integration by Parts: This involves breaking down the integral into two parts and using the integration by parts formula.

What are some common mistakes to avoid when finding the area between 2 functions?

Some common mistakes to avoid when finding the area between 2 functions are:

  • Forgetting to take the absolute value of the difference function before integrating.
  • Forgetting to include the bounds of the interval when evaluating the definite integral.
  • Using the wrong integration method for a particular function.

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