Question

Find the area bounded by the given curves:
y=2x−x2,y=2x−4

Answer 1

`A = [(32)/(3)]`

Explanation:

Given

`y_1 = 2x - x^2`

`y_2 = 2x - 4`

Required

Determine the area bounded by the curves

First, we need to determine their points of intersection

`2x - x^2 = 2x - 4`

Subtract 2x from both sides

`-x^2 = -4`

Multiply through by -1

`x^2 = 4`

Take square root of both sides

`x = 2` or
`x = -2`

This Area is then calculated as thus

`A = \int\limits^a_b {[y_1 - y_2]} \, dx`

Where a = 2 and b = -2

Substitute values for
`y_1` and
`y_2`

`A = \int\limits^a_b {(2x - x^2) - (2x - 4)} \, dx`

Open Brackets

`A = \int\limits^a_b {2x - x^2 - 2x + 4} \, dx`

Collect Like Terms

`A = \int\limits^a_b {2x - 2x- x^2 + 4} \, dx`

`A = \int\limits^a_b {- x^2 + 4} \, dx`

Integrate

`A = [-(x^(3))/(3) +4x](2,-2)`

`A = [-(2^(3))/(3) +4(2)] - [-(-2^(3))/(3) +4(-2)]`

`A = [-(8)/(3) +8] - [-(-8)/(3) -8]`

`A = [(-8+ 24)/(3)] - [(8)/(3) -8]`

`A = [(-8+ 24)/(3)] - [(8-24)/(3)]`

`A = [(16)/(3)] - [(-16)/(3)]`

`A = [(16)/(3)] + [(16)/(3)]`

`A = [(16 + 16)/(3)]`

`A = [(32)/(3)]`

Hence, the Area is:

`A = [(32)/(3)]`