Approximate Area
When finding the area under a curve for a region, it is often easiest to approximate area using a summation series. This approximation is a summation of areas of rectangles. The rectangles can be either left-handed or right-handed and, depending on the concavity, will either overestimate or underestimate the true area.
The Fundamental Theorem of Calculus
The fundamental theorem of Calculus is an important theorem relating antiderivatives and definite integrals in Calculus. The fundamental theorem of Calculus states that if a function f has an antiderivative F, then the definite integral of f from a to b is equal to F(b)-F(a). This theorem is useful for finding the net change, area, or average value of a function over a region.
Definite Integrals as Net Change
Definite integrals can be used to find the net change in a function between two times. The fundamental theorem of Calculus can be restated so that the definite integral of the function's derivative is equal to the net change in the function between two values. Using definite integrals as net change is an accurate way to compute the net change of a quantity.
Average Value of a Function
Calculating the average value of a function over a interval requires using the definite integral. The exact calculation is the definite integral divided by the width of the interval. This calculates the average height of a rectangle which would cover the exact area as under the curve, which is the same as the average value of a function.
The Definite Integral
The definite integral is an important operation in Calculus, which can be used to find the exact area under a curve. The definite integral takes the estimating of approximate areas of rectangles to its limit by using smaller and smaller rectangles, down to an infinitely small size.
Computing Definite Integrals using Substitution
Definite integrals, like their relatives indefinite integrals, can sometimes be solved by using substitution. When computing definite integrals using substitution, the limits of the integral must be modified so they are in terms of the new variable, and not the old one.
Definite Integrals and Area
Definite integrals can be used to find the area under, over, or between curves. If a function is strictly positive, the area between it and the x axis is simply the definite integral. If it is simply negative, the area is -1 times the definite integral. If finding the area between two positive functions, the area is the definite integral of the higher function minus the lower function, or the definite integral of (f(x)-g(x)).
