Question

Why is the area enclosed by \[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\] written as \[4\int_0^a y\,dx\]?

Options

• The ellipse crosses the \[x\]-axis within \[0,a\].

• The ellipse is symmetrical about both the \[x\]-axis and the \[y\]-axis.

• The ellipse lies entirely in the first quadrant.

• The ellipse is bounded by the \[y\]-axis only.

Answer 1

The integral \[\int_0^a y\,dx\] gives the area of region AOBA in the first quadrant. Symmetry about both coordinate axes makes the total enclosed area four times this area.