Definite Integrals Quiz

Study Notes

Definite Integrals

Definite integrals are a type of integral that calculates the area under a curve between two specific points or limits. Unlike indefinite integrals, which produce a function, definite integrals produce a specific numerical value.

Formula and Examples

The definite integral is represented by the symbol ∫f(x) dx from a to b, where f(x) is the function being integrated and a and b are the limits of integration. To solve for definite integrals, follow these steps:

Write down the definite integral with its limits in the form ∫f(x) dx from a to b.
Integrate the function f'(x) the same way you would for an indefinite integral to find f(x).
Evaluate f(x) between the given limits: f(b) - f(a).

For example, to find the definite integral of the function f(x) = x^2 from 1 to 3, we would follow these steps:

∫x^2 dx from 1 to 3
Integrate x^2 with respect to x: (x^3)/3
Evaluate x^3/3 at x = 3 and x = 1: (3^3)/3 - (1^3)/3 = 27/3 - 1/3 = 26/3

Properties of Definite Integrals

Fundamental Theorem of Calculus: The definite integral of a continuous function in a closed interval is equal to the difference between the values of the function at the limits of integration.
Additivity: If the function f(x) is continuous in [a, b] and [b, c], then the definite integral of f(x) from a to c is equal to the sum of the definite integrals from a to b and from b to c.
Linearity: If the function f(x) is continuous in [a, b], and if a and b are in the domain of the function g(x), then the definite integral of f(x)g(x) dx from a to b is equal to the product of the definite integral of f(x) dx from a to b and the definite integral of g(x) dx from a to b.

Riemann Sums

Riemann sums are used to approximate the area under a curve between two points. They consist of a series of rectangles with heights equal to the function at the midpoint of the interval and width equal to the length of the interval. The sum of the areas of these rectangles approaches the definite integral as the number of rectangles increases.

Applications of Definite Integrals

Definite integrals have numerous applications in various fields, including physics, engineering, and economics. They are used to calculate areas, volumes, and net changes over time. For example, in physics, they are used to calculate work, force, and velocity. In economics, they are used to calculate total revenue, total cost, and profit.

In conclusion, definite integrals are a crucial concept in calculus, used to calculate specific numerical values for the area under a curve between two points. They have various applications in different fields and are a fundamental tool for solving real-world problems.