Defining Definite Integrals A definite integral , denoted by \(\int_a^b f(x) dx\), is a specific type of antiderivative of a function defined on the inter...
A definite integral, denoted by (\int_a^b f(x) dx), is a specific type of antiderivative of a function defined on the interval ([a, b]). This means it represents the area under the curve of the function from (a) to (b).
Important properties of definite integrals:
Additivity: (\int_a^b + \int_b^c f(x) dx = \int_a^c f(x) dx)
Constant Factor Rule: (\int k f(x) dx = k \int f(x) dx)
Sum Rule: (\int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx)
Constant Rule: (\int c dx = c x + C) where (C) is a constant
Fundamental Theorem of Calculus: (\int_a^b f(x) dx = f(b) - f(a))
Applications of Definite Integrals:
Calculating areas of shapes (e.g., rectangles, circles, parabolas)
Finding the average value of a function
Determining the total amount of something (e.g., distance, weight, money earned)
Solving optimization problems
Examples:
(\int_0^2 x^2 dx = \left[\frac{x^3}{3} \right]_0^2 = 8)
(\int_1^3 (x^2 - 1) dx = \left[\frac{x^3}{3} - x \right]_1^3 = 6)
(\int_0^10 e^x dx = \lim_{x \to \infty} \left[e^x \right]_0^10 = \infty)
These are just a few basic examples. The properties of definite integrals can be used to solve a wide range of problems involving continuous functions