Integration is one of the fundamental operations in calculus. It allows us to find the area under a curve between two points. The integral notation shows this mathematically, where we integrate function f of x from point a to point b. Geometrically, this represents the signed area between the curve and the x-axis. Integration is also the reverse process of differentiation, helping us find antiderivatives of functions.
Riemann sums provide the foundation for understanding integration. We approximate the area under a curve by dividing it into rectangles. The Riemann sum formula shows how we add up all rectangle areas, where delta x is the width of each rectangle. Starting with just a few wide rectangles, we can see the approximation is rough. But as we increase the number of rectangles, making them narrower, the approximation becomes much more accurate. In the limit as n approaches infinity, this approximation becomes the exact value of the definite integral.
There are two main types of integrals: definite and indefinite. A definite integral has specific bounds from a to b, and it gives us a numerical value representing the area under the curve between those points. The fundamental theorem tells us this equals F of b minus F of a, where F is the antiderivative. An indefinite integral, on the other hand, has no bounds specified. It represents a family of functions, the antiderivatives of the original function, plus an arbitrary constant C. While definite integrals give us specific areas, indefinite integrals give us general function families that differ only by a constant.
The Fundamental Theorem of Calculus is one of the most important results in mathematics. It establishes that differentiation and integration are inverse operations. Part 1 states that if we have an integral from a constant to a variable upper limit, then differentiating this integral with respect to the upper limit gives us back the original function. Part 2 tells us that to evaluate a definite integral, we can find any antiderivative F of our function f, then compute F of b minus F of a. This theorem shows us that the area function, when differentiated, returns the height function, beautifully connecting the geometric concept of area with the algebraic concept of antiderivatives.
Now let's look at the basic integration rules that form the foundation for solving integrals. The power rule states that the integral of x to the n equals x to the n plus 1, divided by n plus 1, plus C, provided n is not equal to negative 1. For exponential functions, the integral of e to the x is simply e to the x plus C. The integral of 1 over x gives us the natural logarithm of the absolute value of x plus C. For trigonometric functions, the integral of sine x is negative cosine x plus C, and the integral of cosine x is sine x plus C. Let's see an example: integrating 3x squared. We factor out the constant 3, apply the power rule to get x cubed over 3, multiply by 3 to get x cubed, and don't forget to add the constant of integration C.