The parabola <em>y</em> = <em>x</em> ² and the line <em>x</em> + <em>y</em> = 12 intersect for
<em>x</em> ² = 12 - <em>x</em>
<em>x</em> ² + <em>x</em> - 12 = 0
(<em>x</em> - 3) (<em>x</em> + 4) = 0
===> <em>x</em> = 3
so you can compute the area by using two integrals,
Then the area you want is
Alternatively, you can subtract the area bounded by <em>y</em> = <em>x</em> ², <em>x</em> + <em>y</em> = 12, and the <em>y</em>-axis in the first quadrant from the area of a triangle with height 12 (the <em>y</em>-intercept of the line) and length 12 (the <em>x</em>-intercept).
Such a triangle has area
1/2 × 12 × 12 = 72
and the area you want to cut away from this is given by a single integral,
The integral has a value of
and so the area of the shaded region is again 72 - 45/2 = 99/2.