Integration - Taking the Integral
Integration is the algebraic method of finding the integral for a function at any point on the graph. Finding the integral of a function with respect to x means finding the area to the x axis from the curve. The integral is usually called the anti-derivative, because integrating is the reverse process of differentiating. The fundamental theorem of calculus shows that antidifferentiation is the same as integration.
The physical concept of the integral is similar to the derivative. For the derivative, the motivation was to find the velocity at any point in time given the position of an object. If we know the velocity of an object at a particular time, the integral will give us the object's position at that time. Just as the derivative gave the instantaneous rate of change, the integral will give the total distance at any given time.
The integral comes from not only trying to find the inverse process of taking the
derivative, but trying to solve the
area problem as well. Just as the process
of differentiation is used to find the slope at any point on the graph, the process
of integration finds the area of the curve up to any point on the graph.
Before integration was developed, we could only really approximate the area of functions by dividing the space into rectangles and adding the areas.
We can approximate the area to the x axis by increasing the number of rectangles under the curve. The area of these rectangles was calculated by multiplying length times width, or y times the change in x. After the area was calculated, the summation of the rectangles would approximate the area. As the number of rectangles gets larger, the better the approximation will be. This is formula for the Riemann Summation, where i is any starting x value and n is the number of rectangles:
This was a tedious process and never gave the exact area for the curve. Luckily, Newton and Leibniz developed the method of integration that enabled them to find the exact area of the curve at any point.
Similar to the way the process of differentiation finds the function of the slope as the distance between two points get infinitesimally small, the process of integration finds the area under the curve as the number of partitions of rectangles under the curve gets infinitely large.
The Definition for the Integral of f(x) from [a,b]
The integral of the function of x from a to b is the sum of the rectangles to the curve at each interval of change in x as the number of rectangles goes to infinity.
The notation on the left side denotes the definite integral of f(x) from a to b. When we calculate the integral from an interval [a,b], we plug a in the integral function and subtract it from b in the integral function:
where F denotes the integrated function. This accurately calculates the area under any continuous function.
The General Power Rule for Integration
To carry out integration, it is important to know the general power rule. It is the exact opposite of the power rule for differentiation.
Let's look at a general function
When we take the integral of the function, we first add 1 to the exponent, and then divide the term by the sum of the exponent and 1.
After we have done this to each term, we add a constant at the end. Recall that taking the derivative of a constant makes it go away, so taking the integral of a function will give us a constant. We label it C because the constant is unknown - it could be any number! Because we can have infinitely as many possible functions for the integral, we call it the indefinite integral.
Let's do an example.
Find the integral of
We start with the first term. We look at the exponent of 2 and increase it by 1, then we divide the term by the resulting exponent of 3.
Then we look at the next term and do the same thing. Since it has an exponent of 1, the resulting exponent will be 2, so we divide the whole term by 2.
The last term has an x value but we just don't see it. We can imagine the last term as -3x0. This is the equivalent to -3(1). If we use the power rule of integration, we add 1 to the exponent to raise it to the first power, and then we divide the term by 1.
All we need to do is add a constant at the end, and we are done.
This power formula for integration works for all values of n except for n = -1 (because we cannot divide by 0). We can take the opposite of the derivative of the logarithmic function to solve these cases.
We should understand
- the Definite Integral as a limit of Riemann sums
- the Definite Integral as a change of quantity over an integral
- how integrals are presented graphically, numerically, and analytically
- how they are interpreted as the position of an object at a given velocity.
To recap, the integral is the function that defines the area under a curve for any given interval. Taking the integral of the derivative of the function will yield the original function. The integral can also tell us the position of an object at any point in time given at least two points of velocity of an object.