# Elliptical Area Under the Cutting Line

Calculus Level 5

An ellipse is centered at $$(5, 6)$$. It has a semi-minor axis of 3 units, and a semi-major axis of 5 units. The semi-minor axis makes an angle of $$60^{\circ}$$ with the $$x$$-axis, and the semi-major axis makes an angle of $$60^{\circ}$$ with the $$y$$-axis, as shown in the figure above. A line whose equation is given by

$y = \dfrac{1}{3} x + 6$

cuts through the ellipse. Find the area of the region bounded by the ellipse that lies under the line. (Shaded area in the figure).