Definite Integrals and Properties A definite integral , denoted by the integral symbol ∫ , is a way of finding the area under the curve of a functi...
A definite integral, denoted by the integral symbol ∫, is a way of finding the area under the curve of a function. Imagine dividing the area under the curve into a finite number of small rectangles and summing the areas of these rectangles. The definite integral represents the sum of the areas of these rectangles, and it gives you the exact value of the area under the curve.
Properties of definite integrals:
Linearity: The definite integral of a sum of functions is equal to the sum of the definite integrals of the individual functions.
Constant factor rule: The definite integral of a constant function is equal to the constant times the definite integral of the independent function.
Sum rule: The definite integral of a function and a constant is equal to the definite integral of the function with the constant multiplied by the original function.
Change of variable: If the independent variable can be expressed in terms of a different variable, we can often change the variable in the definite integral to simplify the calculation.
Examples:
1. Find the definite integral of the function (f(x) = x^2) from (x = 0) to (x = 2).
2. Find the definite integral of the function (f(x) = \frac{1}{x}) from (x = 1) to (x = 2).
3. Find the definite integral of the function (f(x) = \sin(x)) from (x = 0) to (x = \pi).
These examples illustrate the basic concepts and properties of definite integrals and highlight how they can be used to solve various problems in economics