In core three you will be required to know the formulae for integrating a function involving a natural equation. The formulae for the integration of is especially important because if you try to integrate it using the formulae that we learned in core two you will get a zero in the denominator. The formulae are:
The integral of an exponential function to any base is:
Currently we can not integrate directly a composite function, if we need to integrate a composite function we need to perform a linear substitution. To integrate the complex function f[g(x)]', we need to use the following procedure.
Set u = g(x)
Find the derivative of u and solve for dx
In the function replace g(x) with u and dx with the result obtained in step 2.
Integrals are used to find the volume of a shape that is created by rotating a line or set of lines around the x or y axis's. We can only revolve around an axis that is independent. This is also the way to prove the formulae for the volumes of such shapes as cones and spheres. The method that you will learn in this module is known as the disk method. The formulae are:
The procedure is:
Square the function and integrate
Input the highest value.
Input the lower value.
Subtract the result from the higher value from the result of the lower value. The answer has to be positive.
If the curve is bounded by another curve do steps 1 to 5 and then subtract the lower curve from the higher curve. The a is the highest point at which the two curves meet and b is the lowest point at which the two curves meet.
Example One: Area between a Curve and an Axis
Find the volume of the solid obtained by rotating the line around the x-axis and bounded by the line x = 6 and the y - axis.
First we square the function and integrate
Then we input the highest value.
Then we input the lowest value.
Finally we subtract the result from the higher value from the result of the lower value. The answer has to be positive.