ò C is called the constant of integration This is analogous to finding the gradient function. 2

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1 Integration Targets: 1 To use indefinite integrals in the reverse of differentiation 2 To use indefinite integrals to find general and particular solutions 3 To use definite integrals 4 To find areas using definite integrals and the trapezium rule Key Terms or Formulas: Integration is the OPPOSITE of Differentiation Integrate f (x) to find f (x) OR f (x) to find f (x) There are two kinds of integration o Indefinite Integration: produces an expression + C ò C is called the constant of integration This is analogous to finding the gradient function 2 If f ( x) = 3x, then f ( x) = 6x o ò 2 1 Definite Integration: produces a value This is analogous to finding the gradient at a point It gives a value 2 If f ( x) = 3x, then f ( x) = 6x and f ( 5) = 6(5) = 30 Just like your calculator can do!!" [3x2 ] "() = 30 It can also do 3x 2 dx = 7 but you still should show work Area of a Trapezium: A = 05 x height x (base 1 + base 2 ) : Trapezium Rule: A f x dx 3 [y y + y + + y 89 + y 8 ] ; Area is ALWAYS a positive number o However, integrals which model area can produce a negative number when under the x axis Assessments: 2 marks on Short Question 9 marks on Short Test 2 Time Period: 3 weeks wwwmajanmindscom 1

2 Integration Lesson Video Problems Section 1: Reverse Differentiation 1 Integrate: 2x 4 dx 2 Integrate: 9) x 3 dx 3 Integrate: (12 3x + x 2 )dx 3 4 Integrate: m 5 dm 5 Integrate: 2 x 2/3 dx 6 Integrate: 2πdx 7 Integrate: 3 t t + 1 dt 8 Integrate: ( x2 3x 1/2 )dx 9 Integrate: 10 Integrate: 11 Integrate: ("F) G x 1/2 dx " G 9) "9) dx " H 9 "9 dx Section 2: Finding General and Particular Solutions 12 If f I x = 3x 2 2x + 1, find f(x) (This is called the general solution) 13 If!J!" = 6 x3, then find y in terms of x 14 If f II x = ", find f (x) ) 15 If f I x = 3x 2 x and f 0 = 4, find f(x) (This is called the particular solution) 16 Find the equation of the curve whose gradient is given by 2x + 5 and which passes through the point (4, 1) 17 If!M!" = 2x3 + x 2 2x 1 and A = 1 when x = 2, then find the value of A when x = 3 18 Given that f II x = 2x 3 and f I 2 = 4 and f 2 = 0, find f 1 wwwmajanmindscom 2

3 19 The gradient of a curve y that passes through the origin is given by!j!" = 4x x2 Find the coordinates of the points where the curve y cuts the x axis 20 The gradient of a curve is given by!j = ax + b Given that the curve passes through!" (0, 0), (1, 2) and (1, 4), find the values of a and b and give the equation of the curve Section 3: Definite Integration 21 What is Definite Integration? ) 22 Evaluate: (3 5x)dx 23 Evaluate: 24 Evaluate: 9 dx 9 x 2 S 5 x 2 9)" dx 9" R/G 25 When the Limits are the Same: Evaluate: S S xdx 26 When the Limits are the Switched: Compare xdx with xdx 27 When there is a Multiplier: Compare 4xdx with 4 xdx S ; 28 If 2x 3 dx = 6, find the possible values of a U 29 If mx 2 dx = T), find the value of m U 30 If f 2 = 3 and f 1 = 5, find the value of f I x dx Section 4: Finding Areas 31 Area Above the XAxis Find the area bounded by y = x 2 + 1, x = 2, and the axes wwwmajanmindscom 3

Below the XAxis The graph of f(x) is shown Suppose the area of A = 13 units 2 and the area of B = 5 units 2 a) Find the

4 32 Area Below the XAxis Find the area bounded by y = 1 4 x2 1, x = 2, x = 2, and the x axis 33 Area Above and Below the XAxis Find the area of the shaded region which is bounded by y = x 2 2x, x = 2, x = 2, and the x axis 34 Area Above and Below the XAxis The graph of f(x) is shown Suppose the area of A = 13 units 2 and the area of B = 5 units 2 a) Find the total shaded area UW/ 5 W 5 UW/ W b) Find the value of f x dx c) Find the value of f x dx d) Find the value of f x dx wwwmajanmindscom 4

35 Area and a Method of Subtraction Find the area of the shaded region which is bounded by y = x + 2 y = 4, and the y axis 36 Area to the Left of the Curve Find the area of the shaded region

5 35 Area and a Method of Subtraction Find the area of the shaded region which is bounded by y = x + 2 y = 4, and the y axis 36 Area to the Left of the Curve Find the area of the shaded region which is bounded by x = (y 2) 2, y = 4, and the y axis 37 Area to the Left of the Curve Find the area of the shaded region which is bounded by x = 3 y 1, y = 9, and the y axis wwwmajanmindscom 5

by y = 2x + 6 and y = x 2 + 3 40 Area Between a Curve and a Straight Line Find the

6 38 Symmetrical Areas Find the area of the shaded region which is enclosed by y =, y = 1, and y = 3 x2 39 Area Between a Curve and a Straight Line Find the area enclosed by y = 2x + 6 and y = x Area Between a Curve and a Straight Line Find the area enclosed by the x axis, y = 2x, y = x 2 5x + 6, and the x axis wwwmajanmindscom 6

41 Area Between a Curve and a Straight Line The normal to the curve y = x 2 at the point P(1,1) meets the y axis at Q Find the area bounded by the yaxis, the curve y = x 2, and the line PQ 42 Area

7 41 Area Between a Curve and a Straight Line The normal to the curve y = x 2 at the point P(1,1) meets the y axis at Q Find the area bounded by the yaxis, the curve y = x 2, and the line PQ 42 Area Between Two Curves Find the area enclosed between y = 7 x 2 and y = (x 1) The Trapezium Rule Calculate an approximation to the area bounded by y = the Trapezium Rule with 4 strips 16 x 2 and the axes Use wwwmajanmindscom 7

8 44 The Trapezium Rule Calculate an approximation to the area bounded by y = ", x = 1, x = 4 and the x axis Use the Trapezium Rule with 3 strips wwwmajanmindscom 8

9 Answers to Lesson Problems 1 ) x5 + C 2 ) x 2 + C 3 12x U x2 + U x3 + C 4 U [ m8/3 + C 5 6x 1/3 + C 6 2πx + C 7 ] ) t5/2 + 2t 3/2 + C 8 ] x3 6x 1/2 + C 9 ) x5/2 + [ U x3/2 + 8x 1/2 + C 10 x2 + 5x + C 11 U x3 + x2 + x + C 12 f x = x 3 x 2 + x + C 13 y = 6x Ṡ x4 + C 14 f (x) = ) x3/2 + C 15 y = x 3 Ū x3/ y = x 2 + 5x A = 5U ] 18 U ] 19 0,0 and (6,0) 20 a = 6, b = 1, y = 3x 2 x 21 See Video ½ vs vs 6 28 a = 1 29 m = Ū 32 2 Ū a 18 b 8 c 13 d 5 35 [ U 36 [ U Ū 40 1 ) ] wwwmajanmindscom 9

S56 (5.1) Integration.notebook March 09, 2017

S56 (5.1) Integration.notebook March 09, 2017 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