First Quadrant Portion of an Ellipse

Calculus Level 5

An ellipse is centered at \( (3,4) \). It has a semi-minor axis of 3 units, and a semi-major axis of 6 units. Originally, the minor axis is horizontal and the major axis is vertical. The ellipse is rotated counter-clockwise about its center by an angle of \( 30^{\circ} \), resulting in the ellipse shown in the figure above.

Find the area of the portion of the ellipse that lies in the first quadrant (the shaded region). Round your answer to three decimal places.

Hint: Use the special form of Green’s Theorem used for finding areas.

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