Question

Determine the area between the curves by integrating over the x-axis or y-axis.

x=y^2

x=-|y|+12

Answer 1

When
`y\ge0` (above and on the
`x`-axis), we have
`|y|=y`. The parabola
`x=y^2` intersects with
`x=-|y|+12` in this region when

`y^2 = -y + 12 \implies y^2 + y - 12 = (y-3)(y+4) = 0 \implies y=3`

On the other hand, when
`y<0` (below the
`x`-axis, we have
`|y|=-y`, and so the curves intersect when

`y^2 = -(-y) + 12 \implies y^2 - y - 12 = (y - 4)(y+3) = 0 \implies y=-3`

The area between the curves is then given by the definite integral,

`\displaystyle \int_(-3)^3 (-|y| + 12) - y^2 \, dy`

The integrand is symmetric about the
`x`-axis, so the integral is equivalent to

`\displaystyle 2\int_0^3 12 - y - y^2 \, dy = 2 \left(12y - \frac{y^2}2 - \frac{y^3}3\right)\bigg|_0^3 = \boxed{45}`