Question

find the area under the curve y =x^3 from x=0 to x =3 with x axis​

Answer 1

Answer:

we shall play safe and calculate each area separately. We know that the area A

is given by the integral from x = 0 to x = 1 of the curve y = x(x − 1)(x − 2) = x

3 − 3x

2 + 2x;

thus

A =

Z 1

0

y dx

=

Z 1

0

(x

3 − 3x

2 + 2x)dx

=

x

4

4

−

3x

3

3

+

2x

2

2

1

0

=

x

4

4

− x

3 + x

2

1

0

= [ 1

4 − 1 + 1] − [

0

4 − 0 + 0]

=

1

4

.

www.mathcentre.ac.uk 2  c mathcentre 2009

Area B should be given by a similar integral, except that now the limits of integration are from

x = 1 to x = 2:

B =

Z 2

1

y dx

=

Z 2

1

(x

3 − 3x

2 + 2x)dx

=

x

4

4

−

3x

3

3

+

2x

2

2

2

1

=

x

4

4

− x

3 + x

2

2

1

= [ 16

4 − 8 + 4] − [

1

4 − 1 + 1]

= 0 −

1

4

= −

1

4

.

Answer 2

Answer:

Please see the attachment