Exam Techniques

As with the topic on differentiation, the technical integration of functions required for the TMUA is actually fairly basic, with only rational powers of x being included on the specification. (The main exception to this is integration of the modulus function, which requires first splitting the integral into two parts in order to consider negative and positive values of the function separately.) However, what is required, is a good understanding of what integration means; both as the area between a curve and the x-axis, and as the reverse process of differentiation.
 

You should be confident with finding areas between curves and the x-axis for functions that are above or below the x-axis, and especially for functions that are a combination of both. You should understand the difference between calculating the ‘definite integral’ which is obtained by simply substituting in the outer limits, and finding the ‘area’ which might require you to consider the limits of a set of sections of the graph. Remember that area is always positive, whereas an integral can be positive or negative, or even zero!
 

You need to be able to determine the effect of translations, reflections and enlargements on graphs, and how these transformations affect the area(s) between the graph and the coordinate axes. Similarly you should be able to identify relationships between certain areas of graphs that are even, odd, or periodic functions, using the symmetry of these graphs.

Sometimes you might be required to compare the size of integrals of much more complicated functions, such as trigonometric or logarithmic, and here there is no actual integration required, just an ability to sketch the functions and consider the relative size of the integrals being described.

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The idea that differentiation is the reverse process of integration is known as the Fundamental Theorem of Calculus and allows us to combine or separate integrals by adjusting the limits. It also leads to the following useful identity: