Learning Lab

Integration is vital in engineering. It is the key mathematical tool for finding the centre of mass or the surface area of a body.

Integration is also called antidifferentiation. It is the reverse process of differentiation. If you differentiate an expression, you can integrate it to get back to the original. Geometrically, differentiation provides the slope of graph, whereas integration yields the area below.

These integration worksheets will help you to improve your skills in these areas.

• IN1 Antidifferentiation

  How do you antidifferentiate a function? Antidifferentiation (also called integration) is the opposite operation to differentiation. If you have the differentiation of a function, you can then obtain the original function via integration (antidifferentiation).

• IN2 Integration of polynomials

  How do you integrate a polynomial where x is raised to a power? We saw this in the previous section on antidifferentiation. But how do you integrate a linear expression in brackets where the whole bracket is raised to a power? This worksheet will show you how.

• IN3.1 Integration of functions of the form m over (ax+b)

  How do you integrate a logarithm? How do you integrate an exponential function? How do you integrate a trigonometric function?

• IN3.3 Integration of exponential functions

• IN3.4 Integration of trigonometric functions

• IN4 Definite integrals

  Integrating means find the area below a graph. How can you limit this to a certain region? Using definite integrals allows you to find the area below, for a defined section along the horizontal x axis. Read this worksheet to see how.

• IN5 Area under a curve

  An area under a curve might be above the axis (and therefore positive). But sections might also be below the axis (and therefore negative). Read this worksheet to see how to deal with finding the integral for sections of graph which go above and below the x-axis (horizontal axis).

• IN6 Integration by substitution

  An expression that is composed of two functions (say an algebraic expression nested within a trigonometric expression) can be complicated to integrate. You can simplify this by substituting a single pronumeral (say u) to represent one of the functions. The substitution rule for integration is like the chain rule for differentiation: it breaks a complex expression into manageable parts.

• IN7 Integration using partial fractions

  How do you integrate an expression when there is an algebraic expression in the numerator and denominator of a fraction? Integrating using partial fractions helps you to solve this problem. Read this worksheet for several worked examples.

• IN8 Integration by parts

  If you can consider your expression to be a product (i.e. Multiplication x) of two functions, you can integrate this using Integration by parts. This reflects the product rule in differentiation and is applicable to logarithmic, exponential, trigonometric and algebraic functions.

• IN9 Double integrals

  Integrating will find the area between the curve and the x-axis (horizontal axis). We learned in IN4 Definite integrals how to limit this to a section of the x axis. But what if you have an area that is bounded by limits on both the x and the y axes? You will need to integrate with respect to the x axis and then integrate with respect to the y axis as well.

  Double Integrals is the method you will need. Read this worksheet for some examples and exercises. There are linear, exponential and trigonometric functions defining boundaries for shapes on the x-y axes.