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 5 + 2 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
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
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
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 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
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 2 + 3 40 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 Between Two Curves Find the area enclosed between y = 7 x 2 and y = (x 1) 2 + 2 43 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
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
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 x3/2 + 4 16 y = x 2 + 5x 37 17 A = 5U ] 18 U ] 19 0,0 and (6,0) 20 a = 6, b = 1, y = 3x 2 x 21 See Video 22 48 23 ½ 24 138666 25 0 26 15 vs 15 27 6 vs 6 28 a = 1 29 m = 5 30 2 31 4 Ū 32 2 Ū 33 8 34 a 18 b 8 c 13 d 5 35 [ U 36 [ U 37 12 38 29 39 10 Ū 40 1 ) ] 41 42 9 43 1198 44 14583 wwwmajanmindscom 9