S56 (5.1) Integration.notebook March 09, 2017

Size: px

Start display at page:

Download "S56 (5.1) Integration.notebook March 09, 2017"

Claude Taylor
5 years ago
Views:

1 Today we will be learning about integration (indefinite integrals) Integration What would you get if you undo the differentiation? Integration is the reverse process of differentiation. It is sometimes called the anti-derivative. Differentiate 6x 2 x 3 + c 3 x 2 Integrate If we want to integrate a function f(x), we write where dx means with respect to x and what you integrate is called the integral. Integration How to integrate: Given a function f(x) = ax n, then = where n -1 Integration Examples: Integrate the following, giving answers with positive indices C represents the constant of integration and is there to represent any constants that turned to zero when they were differentiated. These types of integrals are called indefinite integrals

2 Integration Examples: Integrate the following, giving answers with positive indices Daily Practice A wind shelter as shown opposite has a back, top and 2 square sides. The total amount of canvas used in the shelter is 96cm 2 and the length of each square side is x metres. (a) If the volume of the shelter is V cm 3, show that V = x(48 - x 2 ) (b) Find the dimensions of the shelter which give a maximum volume. Today we will be continuing to learn to integrate.

3 Integrating a sum or a difference To integrate a sum or a difference, just integrate each term seperately. Examples: Further integrals Further integrals 4. dt 5. dm

4 Q1. Daily Practice Q2.

6 Daily Practice Q1. State the equation of the line perpendicular to the line 7x -2y + 4 = 0, that passes through the point (3, 1) Q2. Write 3x x - 2 in completed square form Today we will be working out definite integrals. Q4. Differentiate (6x - 1) 2 with respect to x x 2

7 Find x 0 Definite integrals Definite integrals have start and end values. They are used for finding the area under a curve. They are of the form where b and a are called the upper and lower limits of integration respectively. Definite integrals Examples: Evaluate

8 Daily Practice Find a for which a > 0 if (1 + 2x) dx = 24 Today we will be calculating the area under a curve using integration. Unit Assessment Assessment Friday 10/3/2017 Topics: Differentiation & Integration Includes questions in context How Integration can be used to find an area y y = f(x) x =b x =a o a b x y o x =a a x =b b y = f(x) x Divide A into a large number of rectangles. The sum of the areas of the rectangles will give an approximate area of A.

9 y y = f(x) y y = f(x) x =a x =b The area of the shaded rectangle is f(x) x h o x =b x =a A(x + h) A(x) a x x + h b x o a x x + h b x This means that A(x + h) - A(x) f(x) x h Therefore A(x + h) - A(x) h f(x) As h gets smaller (closer and closer to zero) i.e. h-> o But then is the same as A'(x) or This means that which then means y 0 4 x y = x that therefore Definite integrals - Area under a curve f(x) dx Definite integrals Examples 1) Write the shaded area as a definite integral. Then work out the shaded area y = 3x + 4 The value of the definite integral represents the area under the curve y = f(x) and the x - axis between the straight lines x = a and x = b y f(x) a b x

10 Definite integrals - 3x - 4 Sheet Daily Practice Q1. Find the point(s) of intersection of the circle x 2 + y 2 + 4x + 2y - 20 = 0 and the line y = 2x + 8 Q2. Calculate Definite integrals when curve is both below & above x - axis Today we will be learning how to find the area between the curve and the x - axis. When given an area that is both below and above the x - axis. Split it into 2 integrals. The area below the x - axis will turn out negative but just ignore the negative signs & add to get the total area.

11 Definite integrals when curve is both below & above x - axis Examples Calculate the shaded area in each graph. Definite integrals when curve is both below & above x - axis - 4x + 3 Q1. M is the point (-3, 0) and N is (6, 6). Find the equation of the line through (4, -1) which is parallel to the line MN. Q2. The first three terms of the recurrence relation Un+1 = pun + q are 14, 12 and 10 respectively. Find the values of p and q. Daily Practice Today we will be finding the area between two graphs.

Area enclosed between two graphs Area enclosed between two graphs To find the area enclosed between two graphs, just take one function away from the other (upper - lower) and then integrate.

12 Area enclosed between two graphs Area enclosed between two graphs To find the area enclosed between two graphs, just take one function away from the other (upper - lower) and then integrate. 1) Write down the shaded y = 19-2x y = 10 - x A = ( f(x) - g(x) ) dx Graph on top - Graph below 2) Calculate the area enclosed(trapped) between the curve y = 4 - x and the line y = 3x. Daily Practice Q1. Differentiate (x - 2) 2 with respect to x Q2. Evaluate (12-3x) dx Today we will be working out differential equations. Q3.

13 Differential Equations Differential equations involve reversing derivatives by integrating them and then finding out what C is. Extra information is needed to evaluate C. Examples: 1. Given, find the equation of the function s = f(t) if s = 65 Differential Equations Examples: 2. The graph y = g(x) passes through the point (3,-1). If express y in terms of x when t = 3 Show that the line y = -3x + 10 is NOT a tangent to the circle with equation x 2 + y 2-8x + 4y - 20 = 0 Integrating Functions of the form (ax + b) n 1. Increase the power by 1 2. Divide by the new power 3. Also divide by the derivative of the expression inside the brackets This rule only works when there are linear expressions inside the bracket. 4. Don't forget C!

14 Integrating Functions of the form (ax + b) n Examples: 1. Find (5x + 4) Integrating Functions of the form (ax + b) n 2. Find Integrating Functions of the form (ax + b) n 3. Find Daily Practice

15 Integrating Trigonometric Functions Once again, think about integrating as being the exact opposite as differentiating. Examples: Integrate (a) 4sinx 0 (b) 2cos2x 0 Integrating Trigonometric Functions Integrating Trigonometric Functions Examples: 1. Find 3cos(6x + )dx Integrating Trigonometric Functions 2. sin(2t - )dt Integrating Trigonometric Functions Examples: 3. Find ( cos(3x) - sin( x+ 1) ) dx to 3 d.p.

Daily Practice 9.3.2017 Integrating Trigonometric Functions Examples: 4.

16 Daily Practice Integrating Trigonometric Functions Examples: 4. A curve for which = 3sin(2x) passes through the point (, 3). Integrating Trigonometric Functions Examples: 5. Find the value of (cos3x - x) dx Find y in terms of x.

S56 (5.3) Further Calculus.notebook March 24, 2016

S56 (5.3) Further Calculus.notebook March 24, 2016 Daily Practice 16.3.2016 Today we will be learning how to differentiate using the Chain Rule. Homework Solutions Video online - please mark 2009 P2 Polynomials HW Online due 22.3.16 We use the Chain Rule