Integration is a fundamental concept in calculus with two main interpretations. First, it's a method for finding the area under a curve. Second, it's the reverse process of differentiation, also known as antidifferentiation. In this example, we're finding the area under the curve f of x equals x squared from x equals zero to x equals three. The result is nine square units.
To find the area under a curve, we use a process called Riemann summation. First, we divide the area into rectangles. Then, we calculate and sum the areas of these rectangles. As we increase the number of rectangles, our approximation becomes more accurate. In the limit as the number of rectangles approaches infinity, we get the exact area. This is the fundamental concept behind integration. The formula shown represents this limiting process mathematically.
Integration can also be understood as antidifferentiation, which is the reverse process of differentiation. If the derivative of capital F of x equals f of x, then capital F of x is an antiderivative of f of x. The indefinite integral of f of x represents the family of all antiderivatives. It's written as the integral of f of x dx equals capital F of x plus C, where C is an arbitrary constant. In this example, the derivative f of x equals 2x is shown in red, and its antiderivative F of x equals x squared is shown in blue. By adding different values of the constant C, we get a family of parallel curves, all with the same derivative.
A definite integral calculates the exact area under a curve between two specific points, a and b. It's written as the integral from a to b of f of x dx. The Fundamental Theorem of Calculus provides a powerful method to evaluate definite integrals. It states that the definite integral equals F of b minus F of a, where F is an antiderivative of f. In this example, we're finding the area under the curve f of x equals x squared between points a and b. As we change the values of a and b, the area and the resulting value of the integral change accordingly. The formula shows that this equals b cubed minus a cubed, all divided by 3.
Let's summarize what we've learned about integration. Integration serves two main purposes: finding areas under curves and reversing the process of differentiation. We can approximate areas using Riemann sums by dividing regions into rectangles and taking the limit as the number of rectangles approaches infinity. Indefinite integrals, written as the integral of f of x dx equals F of x plus C, represent families of antiderivatives. Definite integrals, written as the integral from a to b of f of x dx equals F of b minus F of a, calculate exact areas between specific bounds. Integration is a fundamental concept with applications across physics, engineering, economics, and many other fields where we need to calculate accumulations, totals, or areas.