The area of the region $\{(x, y): 5x^2 ≤ y ≤2x^2 +9\}$ in square units is

$12\sqrt{3}$

Two curves intersect at points whose x coordinates are roots of the equation $5x^2 = 2x^2+9⇒ x=±\sqrt{3}$.

Let A be the area of the shaded region. Then,

$A =\int\limits_{-\sqrt{3}}^{\sqrt{3}} (y_2-y_1) dx = 2\int\limits_{0}^{\sqrt{3}}\left\{(2x^2 +9) -5x^2\right\}dx$

$⇒A=2\int\limits_{0}^{\sqrt{3}}(9-3x^2) dx = 2\left[9x-x^3\right]_{0}^{\sqrt{3}}$

$⇒A=2(9\sqrt{3}-3\sqrt{3})=12\sqrt{3}$ sq. units