Definite itegrals solved equation and examples
A definite integral is a mathematical expression that represents the area under a curve between two specified points (limits). It is denoted by the symbol ∫ and consists of a function inside the integral sign with the limits of integration written on the top and bottom.
The general form for a definite integral is:
∫[a to b] f(x) dx
Where 'f(x)' represents the integrand, 'a' is the lower limit, 'b' is the upper limit, and 'dx' indicates the variable of integration.
To solve a definite integral, you need to find the antiderivative of the integrand and evaluate it at the upper and lower limits, and then subtract the values obtained at the lower limit from the upper limit.
For example, let's solve the definite integral:
∫[0 to 2] x^2 dx
First, we find the antiderivative of x^2, which is (1/3)x^3.
Then, we evaluate it at the upper and lower limits:
(1/3)(2^3) - (1/3)(0^3)
= (1/3)(8) - (1/3)(0)
= 8/3
So, the solution to the definite integral ∫[0 to 2] x^2 dx is 8/3.
Another example:
∫[1 to 5] (2x + 3) dx
We find the antiderivative of (2x + 3), which is x^2 + 3x.
Then, we evaluate it at the upper and lower limits:
(1^2 + 3(1)) - (5^2 + 3(5))
= (1 + 3) - (25 + 15)
= 4 - 40
= -36
So, the solution to the definite integral ∫[1 to 5] (2x + 3) dx is -36.