Further Calculus Applications

Maclaurin series

Some functions can be written as an infinite series in the following form, where the coefficients on powers of are real constants:

We can consider repeatedly differentiating this series:

Then evaluating on to find the constants:

Substituting back into to the original form gives the Maclaurin series for a given function:

Some functions do not have a Maclaurin series, for example is not defined at (nor is any of its derivatives). However, the function does have a Maclaurin series as the function and its derivatives are defined for .

Maclaurin series can be used to approximate integrals of functions that otherwise can't be integrated. By considering the first few terms of the Maclaurin series, a polynomial can be integrated instead of the actual function to approximate the integral.

The following Maclaurin series are given in the formula book:

for all
for
for all
for all
for ,
The values for for which these series are valid is very important.

Improper integrals

Evaluating definite integrals involves finding the indefinite integral, then evaluating the definite integral on the limits:

Improper integrals with infinite limits

One type of improper integral is an integral where one of the limits is infinite:

To evaluate these improper integrals, replace the infinite limit with , find the value of the integral in terms of , then take limits:

The improper integral only has a value if the limit exists and is finite. If the limit does not exist, or is infinite, the improper integral diverges.

Improper integrals with undefined integrands

Another type of improper integral is where the range of integration is finite, but the integrand is not defined at a point within the range of integration. There are two possible cases:

The undefined point is an endpoint of the range of integration
The undefined point is within the range of integration.

If the undefined point is an endpoint of the range of integration, then a similar method can be used to the infinite limits, by replacing the undefined point with , finding the value of the integral in terms of , then taking limits:

If the limit does not exist, or is infinite, the improper integral diverges.

If the undefined point is within the range of integration, then the integral needs to be split into two (for an undefined point ):

If either limit does not exist, or is infinite, the improper integral diverges.

Volumes of revolution

The area between a curve and the -axis from to is given by , if . This area can be revolved about the or axes to create a solid of revolution. The volume of a solid of revolution is the volume of revolution.

When the curve between to is revolved about the -axis, the resulting volume of revolution is given by:

Similarly, when the curve between to is revolved about the -axis, the resulting volume of revolution is given by:

Both of these results can be proven by considering the volume of revolution as being formed of many small cylinders of height or lying along the axis of revolution, each with volume or .

Volumes of revolution can also be formed by revolving the area between two curves. When the area between and between and (where lies above ) is revolved about the -axis, the resulting volume of revolution is given by:

Volumes of revolution can also be calculated for curves defined parametrically. When part of a curve with parametric equation and between points with parameter values and is revolved about one of the coordinate axes, the resulting volume of revolution is given by:

Mean value of a function

The mean value of function between and is given by: