Two Special Definite Integrals
The definite integral of a constant function
1. The definite integral of a constant function:
When integrating a constant function, also known as a function where the value does not change, the result will be the product of the constant and the interval over which it is being integrated.
Mathematically, if we have a constant function f(x) = c and we want to find the definite integral of this function from a to b, where a and b are the limits of integration, the result is:
∫[a to b] c dx = c∫[a to b] dx = c(x)|[a to b] = c(b – a)
For example, if we want to find the definite integral of the function f(x) = 3 from x = 2 to x = 5, we can use the formula mentioned above:
∫[2 to 5] 3 dx = 3(x)|[2 to 5] = 3(5 – 2) = 3(3) = 9.
Therefore, the definite integral of the function f(x) = 3 from x = 2 to x = 5 is equal to 9.
2. The definite integral of the identity function:
The identity function, denoted by f(x) = x, has the property that the integral of the function over any interval is equal to the area under the curve within that interval.
Mathematically, if we want to find the definite integral of the identity function from a to b, we use the following formula:
∫[a to b] x dx = (1/2)x^2|[a to b] = (1/2)(b^2 – a^2)
For example, if we want to find the definite integral of the function f(x) = x from x = 1 to x = 3, we can use the formula mentioned above:
∫[1 to 3] x dx = (1/2)x^2|[1 to 3] = (1/2)(3^2 – 1^2) = (1/2)(9 – 1) = (1/2)(8) = 4.
Therefore, the definite integral of the function f(x) = x from x = 1 to x = 3 is equal to 4.
These are two special cases of definite integrals, but there are many more functions and scenarios to explore within the realm of integration.
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