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

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