Bismillah Adni PreUniversity Cambridge International ALevel Programme Unit

Bismillah. . . Adni Pre-University Cambridge International A-Level Programme Unit P 1: Pure Mathematics 1 (Paper 1) Topic: Integration Chapter 16: Integration

About this chapter • Integration is the reverse process of differentiation.

Learning outcomes • We should:

Finding a function from its derivative Integration: The process of getting from f '(x) to f(x) [Or from dy/dx to y] The reverse process (opposite) of differentiation The symbol is ∫ 2 types of integral: Indefinite (includes an arbitrary constant k) Definite

Differentiation: Reduces the index by 1. n n-1. If f(x) = x , then f '(x) = n x Integration: Increases the index by 1. n If f '(x) = x , then f(x) =

Example 16. 1. 1: The graph of y = f(x) passes through (2, 3) and f '(x) = 6 x 2 – 5 x. Find its equation. To find k, use coordinates (2, 3): When x = 2 and y = 3, k = - 3. Therefore, the equation is:

Example 16. 1. 2: A gardener is digging a plot of land. As he gets tired, he works more slowly. After t minutes, he is digging at a rate of square metres per minute. How long will it take him to dig an area of 40 square metres? Let A square metres be the area he has dug after t minutes. Then, his digging rate is d. A/dt. To find k, use values A = 0 when t = 0 (when he starts to dig): k = 0. Therefore, A = 4√t. To find how long it takes him to dig 40 square metres, substitute A = 40 to get t = 100 minutes.

Calculating areas & the area algorithm Integration is used to calculate areas and volumes. For any function f(x), we can calculate the area bounded by the x-axis, x-axis the graph y = f(x), f(x) and the lines x = a and x = b (the area under the graph from a to b).

To find the area under the graph y = f(x) from x = a to x = b: The 'area under y = f(x) from x = a to x = b' is denoted by: This is called a 'definite integral' (No constant k). The numbers a and b are called the limits (boundaries). The function f(x) is called the integrand The symbol ∫ f(x) dx, dx by itself, without limits (a & b), is used to stand for the 'indefinite integral' integral (with constant k). E. g. :

T o f i n d t h e a r

Example 16. 3. 1: Find the area under x = 2 to x = 5. from

Example 16. 3. 2: Find the area under y = √x from x = 1 to x = 4.

Some properties of definite integrals

Infinite and improper integrals • Infinite integral: – When one of the limits tends to infinity (∞). – E. g. : • Improper integral: – When the integrand is not defined for x = 0. – E. g. :

Example: Find the value of this improper integral

Example: Find the value of this infinite integral

w e e n 2 g r a p h s • To find the area of region bounded by graphs of 2 functions f(x) and g(x), g(x) and 2 lines x = a and x = b: b • or:

Example 16. 6. 1: Show that the graphs of f(x) = x 3 - x 2 - 6 x + 8 and g(x) = x 3 + 2 x 2 - 1 intersect at two points, and find the area enclosed between them. The graphs intersect where f(x) = g(x). x 3 – x 2 – 6 x + 8 = x 3 + 2 x 2 – 1 3 x 2 + 6 x - 9 = 0 3(x + 3)(x - 1) = 0 x = -3 and x = 1 At x = -3, y = -10, so point A (-3, -10). At x = 1, y = 2, so point B (1, 2). There are two points of intersection.

The area between the graphs

i n g ( a x + b ) n n If ∫ g(x) dx = f(x), then Example: Integrate (3 x+1)3.

h e i n t e g r a l o f √ ( 5 -

The End Alhamdulillah. . .