Calculate the Definite Integral: How to Find the Numerical Value of the Area Under a Curve

Definite Integral Definition

A definite integral is a mathematical concept used to calculate the numerical value of the area under a curve between two given points on the x-axis

A definite integral is a mathematical concept used to calculate the numerical value of the area under a curve between two given points on the x-axis. It is basically the integral of a function over a specific interval.

Mathematically, the definite integral is denoted by the symbol ∫ (an elongated “S”) followed by the function to integrate, and enclosed within two points, usually indicated as ‘a’ (lower limit) and ‘b’ (upper limit) representing the interval on the x-axis.

The formula to calculate the definite integral of a function ‘f(x)’ from ‘a’ to ‘b’ is:

∫(from a to b) f(x) dx

It is important to note that the ‘dx’ at the end represents the differential of ‘x’ and signifies that the integration is being done with respect to ‘x’.

To evaluate a definite integral, you need to follow these steps:

1. Start by identifying the given function ‘f(x)’ and the interval (a, b) over which you want to calculate the definite integral.

2. Find the antiderivative of the function ‘f(x)’. The antiderivative of a function is the function that, when differentiated, gives the original function ‘f(x)’. To find the antiderivative, you can use integral rules and techniques.

3. Substitute the upper limit ‘b’ into the antiderivative function and subtract the result of substituting the lower limit ‘a’. The result will be the numerical value of the definite integral.

For example, let’s say we want to calculate the definite integral of the function f(x) = 2x between the interval (1, 3):

∫(from 1 to 3) 2x dx

First, we find the antiderivative of the function 2x, which is x^2. Thus:

∫(from 1 to 3) 2x dx = (x^2)|₁³

Now, we substitute the upper limit ‘3’ into the antiderivative:

(x^2)|₃ = 3^2 = 9

Then, we substitute the lower limit ‘1’ into the antiderivative:

(x^2)|₁ = 1^2 = 1

Finally, we subtract the value at the lower limit from the value at the upper limit:

9 – 1 = 8

Hence, the definite integral of the function f(x) = 2x from 1 to 3 is equal to 8.

This is how you can compute the definite integral using the given interval and the antiderivative of the function.

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