Integration by Parts

When an integral is a product of two functions (ie ), it can be integrated by parts:

(see also the DI method)

Improper Integrals

An integral is improper if either

  • One of its limits is infinity
  • The function is not defined for any point within the interval (bounds inclusive)

To evaluate these integrals, replace the dodgy boundwith a variable , evaluate the integral in terms of the variable, and then take the limit as the variable tends towards the bound.

Where functions are not continuous over the interval, may need to split the function into two integrals. For example, if is not continuous at where , then:

Reduction Formulae

Reduction formulae involve rewriting an integral in terms of itself to get a recurrence relation. They usually involve some variable as well as other variables in the integral (). For example, integrating :

By parts:

Note how the integral is now in terms of itself, but with . This creates a recursive definition that can be expanded to evaluate

Integration by Substitution

Substitution is often useful in solving integrals.

  • Choose a new function
  • Find
  • Substitute in
  • Swap for
  • Put limits in terms of (if appropriate)
  • Solve with respect to u

Choosing a function to substitute depends on the integral, and there are certain patterns to spot which make it easier.


Substituting :

Substituting the limits: The integral becomes:


There are two standard substitutions that can be really useful when integrating trig functions.


The first one:

For example:

Letting :


For example:

Letting :

Standard Forms

Integrals will sometimes be (or can be put into) standard forms which then evaluate directly to inverse trig functions. The full list is given in the data book but:



Trigonometric Identities

Trig identities are often useful in evaluating integrals, for example:

Using :

Integration as a Limit

The area under a curve from is given by: This can be approximated by dividing the area under the curve into a number of rectangles:

For rectangles over the width , the width of each rectangle . The area of the rectangle is therefore given by . The sum of all the rectangles, and therefore total area is:

As , , so:

Volumes of Revolution

For a function rotated 360 degrees about the x axis, consider a disc of width and radius y. The volume is given by . The volume of all slices as is

Therefore the volume of revolution for a function about the x axis is

Volume of revolution about y axis:

Centres of Mass for Planar Objects

The centre of mass is the point through which gravity acts. In 1 dimension:

The sum of the moments about 0 is . The moment of the total mass is . Equating these:

This can be expanded into 2 dimensions:

For the centre of mass of an infinitely thin sheet with uniformly distributed mass, for x-axis consider thin slices of width .

  • Area of slice =
  • Mass of slice =
  • Moment of slice about y-axis =
  • Sum of all moments as =

For the sum of the moments about y axis, take a horizontal slice with width with length

  • Area of slice =
  • Mass of slice =
  • Moments of slice about x-axis =
  • Sum of all moments as =

Note that usually, mass is mass per unit area.


Find centre of mass of plane lamina shown

By symmetry, clearly . For , let be the mass per unit area, and consider a horizontal strip of width .

  • Area of strip is
  • Mass of strip is
  • Moment of one strip about x axis is

Total moment as :

For the total mass , total area of the shape:

So total mass M =

Moments of Inertia for Laminae

The moment of inertia is a measure of how difficult it is to rotate an object. Suppose a lamina is divided into a large number of small elments, each with mass at distance from the origin . The moment of inertia of one element is defined to be . Taking the sum of all moments as

The bounds of the integral should be chosen appropriately such as to include the entire lamina.

  • For a lamina lying in the x-y plane, the moment of inertia about z-axis is the sum of the moments about x and y axes.
  • For an axis parallel to at a distance and both lying in the same plane as the lamina with mass , where passes through the centere of the lamina:


Find the moment of inertia of a thin rectangular plate of mass , length and width about an axis through its centre of gravity which is normal to its plane.

Assuming the plate lies in the x-y axis, the question is asking for the moment about the z-axis. To find this, the moments about both x and y axes are required as . To find :

  • Let the mass per unit area
  • A strip of width at distance from has mass
  • The moment of inertia of the strip is

Taking the limit of the sum of all the strips:

As ,

is identically dervied and equals . Summing the two moments gives:

Lengths of Curves

The length of the arc of a curve between and is given by

Alternatively, for parametrised curves:

Surface Areas of Revolution

Similar to volumes of revolution, the surface area of a function when rotated about the x axis is given by:


The surface are of the parabola between and , when rotated about x axis:

Mean Values of a Function

For a function over the interval

Mean value:

Root mean square value: