Exam Question Starter/Recap

Improper IntegrationLesson 3

Aims:•To know what an improper integral is.•To be able to find the value of an improper integral if it is defined.

What is an improper integral?An integral where:

One limit is undefinedBoth limits are undefinedOr some point in between is undefined

A limit can be undefined if it is equal to ∞, - ∞ or it is an asymptote of the graph

A sketch of the curve can be useful to help decide.

These 3 integrals are improper. Can you explain why?

Improper or not?If you can’t decide, sketch on your calculator and see!

Defined or not?•Sometimes an improper integral can actually be calculated and sometimes not!•It is very hard to tell before you start but a diagram often helps.

When trying to find the integral value we replace the limit that the function is not defined at with a variable, work out the value of the integral in terms of the variable, and then look at what happens as the value of the variable tends to the original value.

Example 1Find, if possible the values of

21 0

1( ) ( )i dx ii xdxx

21

1( )i dxx

0( )ii xdx

Defined or not?

On w/bFind, if possible the values of

33

1( )i dxx

On w/bFind, if possible the values of 0 3( )ii xdx

Example 2Find, if possible the values of

521 1

90 12

1( ) ( ) xi dx ii dxx x

Defined or not?

1

0

1( )i dxx

Defined or not?521 1

91 12

( ) xii dxx

1.

2.

3.

4.

Then do exercise 8C on page 117 questions 1, 3, 5Independent study do revision ex 8 pg 117-118Qu 2,4,6,8,10 compulsory

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