Integration, is the process of anti-differention. In the last segment we saw that differentiating a function gives the gradient (or rate of change) of that function. Integrating a function, on the other hand, gives the area underneath the curve of that function.
The integral sign is . This symbol was chosen by Leibniz because it looks like an elongated S and an integral is a limit of sums. The f(x) is known as the integrand. The dx means that we integrate with respect to x.
An arbitrary constant must be created every time a function is integrated. This is because when a constant is differentiated it becomes zero, so there is no way of telling if such a constant used to exist.
This is the most important rule in calculus it establishes that differentiation and integration are inverse processes. Here is the theorem: Let f be continuous on [a,b].
, F is the antiderivative of f such that F' = f
The first rule states that if we integrate a function and then we differentiate the resultant then we will arrive at the same function. The second part says that we can find a definite integral by subtracting the value of F at the endpoints.
The first part of the Fundamental theorem of calculus establishes that if we differentiate a function and then integrate a function we get a indefinite integral. When we evaluate an indefinite integral the resultant will be a function. For an indefinite integral we use the appropriate rule to get the general antiderivative. If a point on the graph is given we solve for C, to get the complete antiderivative.
The definite integral is used to find the area underneath a curve. The second part of the fundamental theorem of calculus allows to evaluate these integrals, the resultant will be a number. The Definite Integral is denoted as . In the definite integral a is the lower limit and b is the upper limit together they are known as the limits of integration. The limits of integration on which interval to find the area. When we evaluate a definite integral we don't write the +c because they will always cancel out and are liable to cause confusion. When we write this means that we have found the indefinite integral and are going to find the definite integral from b to a.
When we evaluate area underneath curve we need to make sure that the interval over which we are finding the area is not in part or in full below the x-axis. In order to determine this fact we need to find x-intercepts of the function. Then we see if a x-intercept is in the interval. If so we need to determine which part of the interval is below the x-axis. If the whole interval is below the x-axis we take the absolute value of the area .If only part of the interval is under the x-axis we need to break up the integration function into positive and negative parts. For the negative parts we need take the absolute value of the area. For example: If we need to find the area of and on the interval's and the curve is above the x-axis and on the interval the curve is below the x-axis, we would break the integral into these parts: . If you are unable to evaluate the definite integral you will need to use a numerical method to estimate the area.
When we calculate the area between two curves the procedure is very similar as to when we calculate the area bounded by two curves. In the problem the limits of integration will be given. The major difference is that in some cases we need to split the integral up.
Find the places where the 2 curves intercept. .
If the two graphs intercept at a point within the limit of integration, we will need to split the integral into parts. We will use the points of interception as break points.
Decide whether to integrate with respect to x or y. This is easy to do in most cases, see if the function is more easily integrated as y = f(x) or x = f(y).
Determine which curve is above the other curve. If the integral has been split up do this on every interval.
Subtract the top curve from the bottom curve.. If the integral has been split up we add up the parts.
The Trapezium Rule estimates the area under a curve between limits by turning the curve into a set of trapeziums (or strips) and each strip is made out of two ordinates, so there is always one more ordinates than there are strips. The formulae is:
An integral that has either it upper limit going to or is lower limit going to is an improper integral and cannot be computed directly instead we need to find the limit of the function. The improper integrals and is convergent if the limit exists and divergent if the limit does not exist. The rules to calculate a integral going to infinity are: