Area of an ellipse using parametric equations: derivation of area formula, negative area issue.
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3 سال پیش
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In this area of an ellipse using parametric equations video, we quickly derive the area formula for parametric equations by using rectangular area elements. We pay particularly close attention to the effect of direction on the formula: in the derivation of area formula, negative area issue arises if the direction of the curve is reversed. Pay close attention to the sign of x'(t) for the parametric curve, and you will find that sometimes y(t)x'(t) negative and gives negative area contributions. Because geometric area must be positive, this requires a correction to the parametric area integral in the case that the area contributions are negative! We finish the parametric equations derivation of area formula with the caveat that we may need to correct with a minus sign in the case where area counts as negative.
Next, we apply the area formula to the area of an ellipse with semimajor axis "a" and semiminor axis "b". We determine the direction of the parametric curve by substituting several values of t. We investigate the sign of x'(t) and y(t) in each quadrant and find that the area counts as negative in every quadrant. We correct for the negative area issue with a minus sign and write down the area formula. Once we set up the parametric area integral, we are left with a classic trigonometric integral: (sin(x))^2. This requires the use of a trig identity to knock down the power of the trig function. Finally, we guess antiderivatives and evaluate across the limits of integration to obtain the well-known formula for the area of an ellipse: pi*a*b.
Questions or requests? Post your comments below, and I will respond within 24 hours.
In this area of an ellipse using parametric equations video, we quickly derive the area formula for parametric equations by using rectangular area elements. We pay particularly close attention to the effect of direction on the formula: in the derivation of area formula, negative area issue arises if the direction of the curve is reversed. Pay close attention to the sign of x'(t) for the parametric curve, and you will find that sometimes y(t)x'(t) negative and gives negative area contributions. Because geometric area must be positive, this requires a correction to the parametric area integral in the case that the area contributions are negative! We finish the parametric equations derivation of area formula with the caveat that we may need to correct with a minus sign in the case where area counts as negative.
Next, we apply the area formula to the area of an ellipse with semimajor axis "a" and semiminor axis "b". We determine the direction of the parametric curve by substituting several values of t. We investigate the sign of x'(t) and y(t) in each quadrant and find that the area counts as negative in every quadrant. We correct for the negative area issue with a minus sign and write down the area formula. Once we set up the parametric area integral, we are left with a classic trigonometric integral: (sin(x))^2. This requires the use of a trig identity to knock down the power of the trig function. Finally, we guess antiderivatives and evaluate across the limits of integration to obtain the well-known formula for the area of an ellipse: pi*a*b.
3 سال پیش
در تاریخ 1400/01/14 منتشر شده
است.
1,856
بـار بازدید شده