(t) = 3 + 2 cos πt y(t) = 2 + sin πt (a) Find the length of the perimeter of the ellipse. (b) Find the area of the shaded region in the picture.
Added by Cristian M.
Step 1
The equation of an ellipse in standard form is: (x/a)^2 + (y/b)^2 = 1 where a and b are the lengths of the semi-major and semi-minor axes, respectively. To write the equation of the ellipse in standard form, we need to eliminate the parameter t from the Show more…
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Vincenzo Z.
Find the area enclosed by the ellipse $$x=a \cos t, \quad y=b \sin t, \quad 0 \leq t \leq 2 \pi$$
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In the parametrization $c(t)=(a \cos t, b \sin t)$ of an ellipse, $t$ is not an angular parameter unless $a=b$ (in which case, the ellipse is a circle). However, $t$ can be interpreted in terms of area: Show that if $c(t)=(x, y)$, then $t=(2 / a b) A$, where $A$ is the area of the shaded region in Figure 29 . Hint: Use Eq. (9).
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