1. Integration Please choose a question to attempt from the following: 12345

2. y = x 2 - 8x + 18 x = 3 x = k Show that the shaded area is given by 1 / 3 k 3 – 4k 2 + 18k - 27 INTEGRATION : Question 1 Go to full solution Go to Marker’s Comments Go to Main Menu Reveal answer only The diagram below shows the curve y = x 2 - 8x + 18 and the lines x = 3 and x = k.

3. y = x 2 - 8x + 18 x = 3 x = k Show that the shaded area is given by 1 / 3 k 3 – 4k 2 + 18k - 27 INTEGRATION : Question 1 Go to full solution Go to Marker’s Comments Go to Main Menu Reveal answer only Area = (x 2 - 8x + 18) dx  3 k = 1 / 3 k 3 – 4k 2 + 18k – 27 as required.

4. Markers Comments Begin Solution Continue Solution Question 1 Back to Home The diagram shows the curve y = x 2 - 8x + 18 and the lines x = 3 and x = k. Show that the shaded area is given by 1 / 3 k 3 – 4k 2 + 18k - 27 Area = (x 2 - 8x + 18) dx  3 k = x 3 - 8x 2 + 18x [][] 3 2 k 3 = 1 / 3 x 3 – 4x 2 + 18x [][] k 3 = ( 1 / 3 k 3 – 4k 2 + 18k) – (( 1 / 3 X 27) – (4 X 9) + 54) = 1 / 3 k 3 – 4k 2 + 18k – 27 as required.

5. Markers Comments Back to Home Next Comment Area = (x 2 - 8x + 18) dx  3 k = x 3 - 8x 2 + 18x [][] 3 2 k 3 = 1 / 3 x 3 – 4x 2 + 18x [][] k 3 = ( 1 / 3 k 3 – 4k 2 + 18k) – (( 1 / 3 X 27) – (4 X 9) + 54) = 1 / 3 k 3 – 4k 2 + 18k – 27 as required. Learn result can be used to find the enclosed area shown: a b f(x)

6. Markers Comments Back to Home Next Comment Area = (x 2 - 8x + 18) dx  3 k = x 3 - 8x 2 + 18x [][] 3 2 k 3 = 1 / 3 x 3 – 4x 2 + 18x [][] k 3 = ( 1 / 3 k 3 – 4k 2 + 18k) – (( 1 / 3 X 27) – (4 X 9) + 54) = 1 / 3 k 3 – 4k 2 + 18k – 27 as required. Learn result for integration: “Add 1 to the power and divide by the new power.”

7. INTEGRATION : Question 2 Go to full solution Go to Marker’s Comments Go to Main Menu Reveal answer only Given that dy / dx = 12x 2 – 6x and the curve y = f(x) passes through the point (2,15) then find the equation of the curve y = f(x).

8. INTEGRATION : Question 2 Go to full solution Go to Marker’s Comments Go to Main Menu Reveal answer only Given that dy / dx = 12x 2 – 6x and the curve y = f(x) passes through the point (2,15) then find the equation of the curve y = f(x). Equation of curve is y = 4x 3 – 3x 2 - 5

9. Markers Comments Begin Solution Continue Solution Question 2 Back to Home Given that dy / dx = 12x 2 – 6x and the curve y = f(x) passes through the point (2,15) then find the equation of the curve y = f(x). dy / dx = 12x 2 – 6x So = 12x 3 – 6x 2 + C 3 2 = 4x 3 – 3x 2 + C Substituting (2,15) into y = 4x 3 – 3x 2 + C We get 15 = (4 X 8) – (3 X 4) + C So C + 20 = 15 ie C = -5 Equation of curve is y = 4x 3 – 3x 2 - 5

10. Markers Comments Back to Home Next Comment dy / dx = 12x 2 – 6x So = 12x 3 – 6x 2 + C 3 2 = 4x 3 – 3x 2 + C Substituting (2,15) into y = 4x 3 – 3x 2 + C We get 15 = (4 X 8) – (3 X 4) + C So C + 20 = 15 ie C = -5 Equation of curve is y = 4x 3 – 3x 2 - 5 Learn the result that integration undoes differentiation: i.e.given Learn result for integration: “Add 1 to the power and divide by the new power”.

11. Markers Comments Back to Home Next Comment dy / dx = 12x 2 – 6x So = 12x 3 – 6x 2 + C 3 2 = 4x 3 – 3x 2 + C Substituting (2,15) into y = 4x 3 – 3x 2 + C We get 15 = (4 X 8) – (3 X 4) + C So C + 20 = 15 ie C = -5 Equation of curve is y = 4x 3 – 3x 2 - 5 Do not forget the constant of integration!!!

12. INTEGRATION : Question 3 Go to full solution Go to Marker’s Comments Go to Main Menu Reveal answer only Findx 2 - 4 2x  x dx 

13. INTEGRATION : Question 3 Go to full solution Go to Marker’s Comments Go to Main Menu Reveal answer only Findx 2 - 4 2x  x dx  =x  x + 4 + C 3  x

14. Markers Comments Begin Solution Continue Solution Question 3 Back to Home Findx 2 - 4 2x  x dx  x 2 - 4 2x  x dx  = x 2 - 4 2x 3 / 2 2x 3 / 2 dx  = 1 / 2 x 1 / 2 - 2x -3 / 2 dx  = 2 / 3 X 1 / 2 x 3 / 2 - (-2) X 2x -1 / 2 + C = 1 / 3 x 3 / 2 + 4x -1 / 2 + C =x  x + 4 + C 3  x

15. Markers Comments Back to Home Next Comment x 2 - 4 2x  x dx  = x 2 - 4 2x 3 / 2 2x 3 / 2 dx  = 1 / 2 x 1 / 2 - 2x -3 / 2 dx  = 2 / 3 X 1 / 2 x 3 / 2 - (-2) X 2x -1 / 2 + C = 1 / 3 x 3 / 2 + 4x -1 / 2 + C =x  x + 4 + C 3  x Prepare expression by: 1 Dividing out the fraction. 2 Applying the laws of indices. Learn result for integration: Add 1 to the power and divide by the new power. Do not forget the constant of integration.

16. INTEGRATION : Question 4 Go to full solution Go to Marker’s Comments Go to Main Menu Reveal answer only 1 2  ( ) Evaluatex 2 - 2 2 dx x

17. INTEGRATION : Question 4 Go to full solution Go to Marker’s Comments Go to Main Menu Reveal answer only 1 2  ( ) Evaluatex 2 - 2 2 dx x = 2 1 / 5

18. Markers Comments Begin Solution Continue Solution Question 4 Back to Home 1 2  ( ) Evaluatex 2 - 2 2 dx x = 2 1 / 5 1 2  ( ) x 2 - 2 2 dx x ( ) = x 4 - 4x + 4 dx  x2x2 1 2 ( ) = x 4 - 4x + 4x -2 dx  1 2 [ ] = x 5 - 4x 2 + 4x -1 5 2 -1 1 2 = x 5 - 2x 2 - 4 5 x [ ] 2 1 = ( 32 / 5 - 8 - 2) - ( 1 / 5 - 2 - 4)

19. Markers Comments Back to Home Next Comment = 2 1 / 5 1 2  ( ) x 2 - 2 2 dx x ( ) = x 4 - 4x + 4 dx  x2x2 1 2 ( ) = x 4 - 4x + 4x -2 dx  1 2 [ ] = x 5 - 4x 2 + 4x -1 5 2 -1 1 2 = x 5 - 2x 2 - 4 5 x [ ] 2 1 = ( 32 / 5 - 8 - 2) - ( 1 / 5 - 2 - 4) Prepare expression by: 1 Expanding the bracket 2 Applying the laws of indices. Learn result for integration: “Add 1 to the power and divide by the new power”. When applying limits show substitution clearly.

20. (a) Find the coordinates of A and B. (b)Hence find the shaded area between the curves. y = -x 2 + 8x - 10 y = x A B INTEGRATION : Question 5 Go to full solution Go to Marker’s Comms Go to Main Menu Reveal answer only The diagram below shows the parabola y = -x 2 + 8x - 10 and the line y = x. They meet at the points A and B.

21. (a) Find the coordinates of A and B. (b)Hence find the shaded area between the curves. y = -x 2 + 8x - 10 y = x A B INTEGRATION : Question 5 Go to full solution Go to Marker’s Comms Go to Main Menu Reveal answer only The diagram below shows the parabola y = -x 2 + 8x - 10 and the line y = x. They meet at the points A and B. A is (2,2) and B is (5,5). = 4 1 / 2 units 2

22. Markers Comments Begin Solution Continue Solution Question 5 Back to Home The diagram shows the parabola y = -x 2 + 8x - 10 and the line y = x. They meet at the points A and B. (a) Find the coordinates of A and B. (b)Hence find the shaded area between the curves. (a)Line & curve meet when y = x and y = -x 2 + 8x - 10. Sox = -x 2 + 8x - 10 orx 2 - 7x + 10 = 0 ie(x – 2)(x – 5) = 0 iex = 2 or x = 5 Since points lie on y = x then A is (2,2) and B is (5,5).

23. Markers Comments Begin Solution Continue Solution Question 5 Back to Home The diagram shows the parabola y = -x 2 + 8x - 10 and the line y = x. They meet at the points A and B. (a) Find the coordinates of A and B. (b)Hence find the shaded area between the curves. A is (2,2) and B is (5,5). (b) Curve is above line between limits so Shaded area = (-x 2 + 8x – 10 - x) dx  2 5 = (-x 2 + 7x – 10) dx  2 5 = -x 3 + 7x 2 - 10x 3 2 [] 5 2 = ( -125 / 3 + 175 / 2 – 50) – ( -8 / 3 +14 – 20) = 4 1 / 2 units 2

24. Markers Comments Back to Home Next Comment (a)Line & curve meet when y = x and y = -x 2 + 8x - 10. Sox = -x 2 + 8x - 10 orx 2 - 7x + 10 = 0 ie(x – 2)(x – 5) = 0 iex = 2 or x = 5 Since points lie on y = x then A is (2,2) and B is (5,5). At intersection of line and curve y 1 = y 2 Terms to the left, simplify and factorise.

25. Markers Comments Back to Home Next Comment (b) Curve is above line between limits so Shaded area = (-x 2 + 8x – 10 - x) dx  2 5 = (-x 2 + 7x – 10) dx  2 5 = -x 3 + 7x 2 - 10x 3 2 [] 5 2 = ( -125 / 3 + 175 / 2 – 50) – ( -8 / 3 +14 – 20) = 4 1 / 2 units 2 Learn result can be used to find the enclosed area shown: a b upper curve y1y1 area y2y2 lower curve