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