Question

Find the area bounded by the curve `y=2x^2-6x` and `y=-x^2+9`?

Answer 1

`"Area" = 32`

Explanation:

First, it is a good idea to graph the given curves:

We need to evaluate the area bounded by these curves, i.e find the area in between them.

That's the area between their points of intersection, so we need to find that: their points of intersection:

`Rightarrow 2x^(2) - 6x = - x^(2) + 9`

`Rightarrow 3x^(2) - 6x - 9 = 0`

`Rightarrow x^(2) - 2x - 3 = 0`

`Rightarrow x^(2) + x - 3x - 3 = 0`

`Rightarrow x (x + 1) - 3 (x + 1) = 0`

`Rightarrow (x + 1)(x - 3) = 0`

`therefore x = - 1, 3`

We don't really need to find the points of intersection; the `x`-intercepts will suffice.

This is because the curves are already specified in terms of `y`.

Now we can start evaluating the definite integrals of these curves in the interval `[- 1, 3]`:

`Rightarrow int_(- 1)^(3)` `((- x^(2) + 9) - (2x^(2) - 6x))` `dx`

`= int_(- 1)^(3)` `(- 3x^(2) + 6x + 9)` `dx`

`= |- x^(3) + 3x^(2) + 9x|_(- 1)^(3)`

`= (- (3)^(3) + 3(3)^(2) + 9(3)) - (- (- 1)^(3) + 3(- 1)^(2) + 9(- 1))`

`= (- 27 + 27 + 27) - (1 + 3 - 9)`

`= (27) - (- 5)`

`= 32`

Therefore, the area bounded by the curves is `32`.