2 x 3 ∫ x 2 ( ) 1 x - WHEC MATHS Mr Stewart -...

Integration

1. Calculate (a) ∫ x2(x – 5) dx (b)

24x 3

x1

+∫ dx (c) ( )22

1 x

2

2

x ∫− + dx

y = x2 – 6x

2. Calculate the shaded area in the diagram

opposite.

3. The diagram shows part of the graph of

y = 7x – 10 – x2.

Calculate the shaded area.

y = 2x2 + 2x – 20

4. Find the area between the curves shown.

y = 40 + 5x – x2

x

y

x

y

2 5

x

y

5. The diagram opposite illustrates the graph

of y = f(x) where f(x) = 2x3 – x

2 – 8x + 4.

(a) Show that x – 2 is a factor of f(x) and

hence fully factorise f(x).

(b) Calculate the shaded area.

6. The diagram shows part of y = f(x).

(a) Find a formula for f(x).

(b) Calculate the area enclosed by f(x) and

the x-axis.

7. (a) Find the equation of the parabola opposite.

(b)..Hence calculate the shaded area between

this parabola and the line y = 2x.

y = x2 – px + 12

8.In the diagram opposite the area shown

is 60. The curve has equation

y = x2 – px + 12.

Calculate the value of p.

x

y

2

x

y

-1 1

4

x

y

(4,32)

y = 2x

8O

x

y

6

12

9. The diagram opposite shows the curve

y = x3 +2x

2 – 5x – 5 and the line AB.

The line AB is a tangent to the curve

at the point A(-2,5).

(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 by 28 metres high.

The roof of the tunnel is in the form of a

parabola with equation y = 2

61 x - 24 .

(a) Find the coordinates of A and B.


11. f /(x) = 3x

2 – 4x + 6 and f(2) = 17.

Find a formula for f(x).

12. f /(x) =

x

x- 2x 23

and f(6) = 100.


13. f /(x) = 4x(x

2 – 1) and f(-1) = 2. Find a formula for f(x).

14. The graph of y = g(x) passes through the point (3,-1).

If 2

2

x

1 - 3x

dx

dy= , express y in terms of x.

f(x)

15. The graphs of y = f(x) and y = g(x)

intersect at the point A on the y-axis. A

If g(x) = 4x + 2 and f /(x) = 2x – 6,

find f(x).

g(x)

x

yA(-2,5)

B

X

Y

18 m 18 m

28 m

A B

x

y

Integration 2

1. Find ∫ (2x2 – 2)(x

2 + 1) dx.

2. Given ∫ ( )6- dx

1 -2x

102

= , find p.

3. The diagram shows part of the graph of y = 3x2 – x

3.


4. (a) f(x) crosses the x-axis at (1,0) and (5,0) and has a

maximum turning point at (3,8).


(b) Calculate the area under the curve.

y = x3 – x

2 – 6x – 2

5. The diagram shows the curve y = x3 – x

2 – 6x – 2

and the straight line AB. This line is a tangent to

the curve at the point A(1,-8).

(a) Find the equation of this tangent at A.

(b) Find the coordinates of B.

(c) Calculate the shaded area.

−1 1 2 3 4

−2

−1

1

2

3

4

y

y

1 5

(3,8)

x

y

B

A

6. The diagram shows the graph of y = x2 – 5x + 4.


f(x)

7. The diagram shows the graphs of f(x) = x2 – 4 and

g(x) = 4 – x2.

(a) Find the coordinates of A and B. A B


g(x)

8. The graph shows the line y = 2x + 8

and the curve y = x2 + 3x – 4.

Calculate the area between the line

and the curve.

x

y

x

y

x

y

9. The diagram shows the cubic function

y = x3 and the line y = x.


y = x

y = x3

y = 21 x

2 + 4

10. Shown is part of the parabola

y = 21 x

2 + 4.


x

y

x

y

-2 2

The Area under a Curve

1. The diagram opposite shows the graph of

y = x2 – 5x.


2. The diagram shows the graph of

y = 4x – x2.

Calculate the area between the

curve and the x-axis.

3. The diagram shows part of the graph

of y = 6x + 2x2.

(a) Find the coordinates of A.


4. The dagram shows part of the graph of

y = 2x2 – 18.

(a) Calculate the coordinates of P and Q.

(b) Find the shaded area.


y = 3 – 3x2.



y = x2 – 2x – 12.



y = 10 – 4x – 3x2



y = 8 – 2x – x2.



9. The diagram opposite shows part of the

graph of y = x3 – 3x

2 + 2x.

(a) Find the coordinates of P and Q.



y = x3 – 3x

2 – 10x.




y = x3 – 4x

2 – 7x + 10.




y = x4 – 5x

2 + 4.


Area Between two Curves


y = 4x – x2 and the line y = 3.



2. The curves with equations y = x2 and

y = 2x2 – 25 intersect at P and Q.

Calculate the area enclosed between

the curves.


y = 7x – 2x2 and the line y = 3x.


4. The curves with equations y = 2x2 – 6

and y = 10 – 2x2 intersect at K and L.

Calculate the area enclosed by these two

curves.

5. The diagram opposite shows part of

the curves y = x3 + x

2 and

y = 2x2 + 2x.


6. The curve y = x(x – 3)(x + 3) and

the line y = 7x intersect at the points

(0,0), (-4,-28) and (4,28).

Calculate the area enclosed by the

curve and the line.

7. The parabolas y = x2 – 4x + 8 and

y = 8 + 4x – x2 intersect at A and B.



8. The diagram shows parts of the curves

y = x3 – 1 and y = x

2 – 1.


9. The curve y = x3 – x

2 – 7x + 5 and the

line y = 2x – 4 are shown opposite.

(a) B has coordinates (1,-2). Find the

coordinates of A and C.

(b) Hence calculate the shaded area.

10. The diagram shows the line y = 3x – 5 and

the curve y = x3 – 5x

2 – 5x + 7.

(a) Find the coordinates of P and Q.


11. The diagram opposite shows an area

enclosed by 3 curves:

y = x(x + 3), 2x

4 y = and 2x

4

1 - x y =

(a) P and Q have coordinates (p,4) and (q,1).

Find the values of p and q.


Differential Equations

1. f / (x) = 6x

2 – 4x. Given f(2) = 10 find a formula for f(x).

2. f / (x) = 10x – 2x

3. Given f(2) = 0 find a formula for f(x).

3. dx

dy= 3 + 6x – x

2. When x = 3, y = 10. Find a formula for y.

4. dx

dy= 6x –

2x

5. Find a formula for the curve y given it passes through the

point (1,6).

5. dx

dy= 6x. - x3 Find a formula for y given the curve passes through (4,-30).

6. The gradient of the tangent to a curve is given by f / (x) = 6x

2 – 4. If the curve

passes through the point (2,7), find its equation.

7. The gradient of the tangent to a curve is given by dx

dy = 1

x

2+ . If the curve

passes through the point (9,10), find its equation.

8. f /(x) = 3x

2 – 4x + 6 and f(2) = 17.


9. f /(x) =

x

x- 2x 23

and f(6) = 100.


10. f /(x) = 4x(x

2 – 1) and f(-1) = 2. Find a formula for f(x).

11. The graph of y = g(x) passes through the point (3,-1).

If 2

2

x

1 - 3x

dx

dy= , express y in terms of x.

f(x)

12. The graphs of y = f(x) and y = g(x) A

intersect at the point A on the y-axis.

If g(x) = 4x + 2 and f /(x) = 2x – 6,

find f(x). x

y

2 x 3 ∫ x 2 ( ) 1 x - WHEC MATHS Mr Stewart -...

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