# Graph the line v = x and the parabola v = 6-x^2. Find the area of the region bounded by each.

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Expert Answers

hala718 | Certified Educator

Given the graphs:

y= x

y= `6-x^2`

We need to find the area bounded by the graphs.

First, we will find the intersection points between the line and the parabola.

`==gt x = 6-x^2 `

`==gt x^2 + x - 6 = 0`

==> (x+3)(x-2) = 0

==> x= -3 , x= 2

Then, we will find the area bounded by the graphs and between the lines x= -3 and x= 2.

We know that the area between the graphs is the integral of the curve (`6-x^2` ) minus the integral of the line (`y=x` ).

`==gt Area = int_-3^2 (6-x^2) dx - int_-3^2 x dx`

`==gt Area = 6x - x^3/3 - x^2/2`

`==gt Area = 6(-3) - (-3^3)/3 - (-3^2)/2 ` `- ( 6(2) - 2^3/3 - 2^2/2)`

`==gt Area = -18 + 9 - 9/2 - 12 + 8/3 + 2 `

`==gt Area = 20.83`