( ) 2. Integration. 1. Calculate (a) x2 (x 5) dx (b) y = x 2 6x. 2. Calculate the shaded area in the diagram opposite.

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1 Integration 1. Calculate (a) ( 5) d (b) d (c) ( ) + d 1 = 6. Calculate the shaded area in the diagram opposite. 3. The diagram shows part of the graph of = = Find the area between the curves shown. =

2 5. The diagram opposite illustrates the graph of = f() where f() = (a) Show that is a factor of f() and hence full factorise f(). (b) 6. The diagram shows part of = f(). 4 (a) Find a formula for f(). (b) Calculate the area enclosed b f() and the -ais (4,3) 7. (a) Find the equation of the parabola opposite. (b)..hence calculate the shaded area between this parabola and the line =. = O 8 8.In the diagram opposite the area shown is 60. The curve has equation = p + 1. Calculate the value of p. 1 = p + 1 6

3 9. The diagram opposite shows the curve = and the line AB. The line AB is a tangent to the curve at the point A(-,5). A(-,5) B (a) Find the equation of the tangent AB. (b) Hence find the coordinates of B. (c) Calculate the shaded area between the curve and the line. 10. The diagram shows a tunnel 36 metres wide b 8 metres high. The roof of the tunnel is in the form of a parabola with equation = 4-1. (a) Find the coordinates of A and B. (b) 6 8 m Y 18 m 18 m A B X 11. f / () = and f() = 17. Find a formula for f(). 1. f / 3 - () = and f(6) = 100. Find a formula for f(). 13. f / () = 4( 1) and f(-1) =. Find a formula for f(). 14. The graph of = g() passes through the point (3,-1). If d 1 = 3 -, epress in terms of. d f() 15. The graphs of = f() and = g() intersect at the point A on the -ais. If g() = 4 + and f / () = 6, find f(). g() A

4 Integration 1. Find ( )( + 1) d. 10. Given ( -1) d = - 6, find p. 3. The diagram shows part of the graph of = (3,8) 4. (a) f() crosses the -ais at (1,0) and (5,0) and has a maimum turning point at (3,8). Find a formula for f(). (b) Calculate the area under the curve The diagram shows the curve = 3 6 and the straight line AB. This line is a tangent to the curve at the point A(1,-8). B = 3 6 (a) Find the equation of this tangent at A. (b) Find the coordinates of B. (c) A

5 6. The diagram shows the graph of = f() 7. The diagram shows the graphs of f() = 4 and g() = 4. (a) Find the coordinates of A and B. A B (b) g() 8. The graph shows the line = + 8 and the curve = Calculate the area between the line and the curve.

6 9. The diagram shows the cubic function = 3 and the line =. = = 3 = Shown is part of the parabola =

7 The Area under a Curve 1. The diagram opposite shows the graph of = 5.. The diagram shows the graph of = 4. Calculate the area between the curve and the -ais. 3. The diagram shows part of the graph of = 6 +. (a) Find the coordinates of A. (b) 4. The dagram shows part of the graph of = 18. (a) Calculate the coordinates of P and Q. (b) Find the shaded area.

8 5. The diagram shows part of the graph of = The diagram shows the graph of = The diagram shows part of the graph of = The diagram shows the graph of = 8. (a) Find the coordinates of A and B. (b)

9 9. The diagram opposite shows part of the graph of = (a) Find the coordinates of P and Q. (b) 10. The diagram shows the graph of = (a) Find the coordinates of A and B. (b) 11. The diagram shows the graph of = (a) Find the coordinates of A and B. (b) 1. The diagram shows the graph of =

10 Area Between two Curves 1. The diagram opposite shows the curve = 4 and the line = 3. (a) Find the coordinates of A and B. (b). The curves with equations = and = 5 intersect at P and Q. Calculate the area enclosed between the curves. 3. The diagram opposite shows the curve = 7 and the line = The curves with equations = 6 and = 10 intersect at K and L. Calculate the area enclosed b these two curves.

11 5. The diagram opposite shows part of the curves = 3 + and = The curve = ( 3)( + 3) and the line = 7 intersect at the points (0,0), (-4,-8) and (4,8). Calculate the area enclosed b the curve and the line. 7. The parabolas = and = intersect at A and B. (a) Find the coordinates of A and B. (b)

12 8. The diagram shows parts of the curves = 3 1 and = The curve = and the line = 4 are shown opposite. (a) B has coordinates (1,-). Find the coordinates of A and C. (b) Hence calculate the shaded area. 10. The diagram shows the line = 3 5 and the curve = (a) Find the coordinates of P and Q. (b) 11. The diagram opposite shows an area enclosed b 3 curves: 4 = ( + 3), = and 1 = - 4 (a) P and Q have coordinates (p,4) and (q,1). Find the values of p and q. (b)

13 Differential Equations 1. f / () = 6 4. Given f() = 10 find a formula for f().. f / () = Given f() = 0 find a formula for f(). d 3. = When = 3, = 10. Find a formula for. d d 5 4. = 6 d. Find a formula for the curve given it passes through the point (1,6). 5. d = 3-6. Find a formula for given the curve passes through (4,-30). d 6. The gradient of the tangent to a curve is given b f / () = 6 4. If the curve passes through the point (,7), find its equation. d 7. The gradient of the tangent to a curve is given b = + 1. If the curve d passes through the point (9,10), find its equation. 8. f / () = and f() = 17. Find a formula for f(). 9. f / 3 - () = and f(6) = 100. Find a formula for f(). 10. f / () = 4( 1) and f(-1) =. Find a formula for f(). 11. The graph of = g() passes through the point (3,-1). If d 1 = 3 -, epress in terms of. d f() 1. The graphs of = f() and = g() A intersect at the point A on the -ais. If g() = 4 + and f / () = 6, find f().

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