© Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules 22a: Integrating the Simple Functions

Integrating the Simple Functions "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Module C3 OCR

Integrating the Simple Functions Before we look again at integration we need to remind ourselves how to differentiate the simple functions. What goes here?

Integrating the Simple Functions We also need to know that multiplying constants just “tag along” and that terms like the above can be differentiated independently when they appear in sums and differences. e.g.

Integrating the Simple Functions Indefinite integration is just the reverse of differentiation, so, reading the differentiation table from right to left, we get: so we use We don’t want to remember the formula with,

Integrating the Simple Functions Indefinite integration is just the reverse of differentiation, so, reading the differentiation table from right to left, we get: is only defined for x > 0, so we write which means negative signs are ignored.

Integrating the Simple Functions SUMMARY Which function is “missing” from the l.h.s. and why?

Integrating the Simple Functions SUMMARY We can’t yet integrate since we haven’t found a function that differentiates to give.

Integrating the Simple Functions We will next practise using the integrals of the simple functions by evaluating some definite integrals and finding some areas. Reminder: To find we write If, by mistake, we do a similar thing with ( forgetting that it gives ), we get. Why is this impossible? Then, using the 1 st rule ANS: We can’t divide by zero.

Integrating the Simple Functions e.g. 1. Evaluate the following integrals: Solutions: (a) Be careful here... Substituting x = 0 does not give 0. (a) (b)

Integrating the Simple Functions Solutions: (a) The integral gives the shaded area. We need to remember that e.g. 1. Evaluate the following integrals: (a) (b)

Integrating the Simple Functions (b) Since the limits are positive, the mod sign makes no difference so we can now omit it.

Integrating the Simple Functions Exercises Evaluate the following integrals: In each case sketch a graph and briefly explain how your answer relates to area.

Integrating the Simple Functions Solutions: 1. The areas above and below the axis are equal, but the integral for the area below is negative.

Integrating the Simple Functions The area is above the axis, so the integral gives the entire area. 2.

Integrating the Simple Functions

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Integrating the Simple Functions SUMMARY

Integrating the Simple Functions Solutions: (a) The integral gives the shaded area. We need to remember that e.g. 1. Evaluate the following integrals: (a) (b)

Integrating the Simple Functions Since the limits are positive, the mod sign makes no difference so we can now omit it. N.B. When working out definite integrals we need to remember that some functions don’t give 0 when x = 0. In particular, (b)