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3.15 Table for Simpson’s rule

We use Simpson’s rule in the following table.

θ\theta00π4\frac{\pi}{4}π2\frac{\pi}{2}sin⁡θ\sin\theta1-1625⁢sin2⁡θ\sqrt{1-\frac{16}{25}\sin^{2}\theta}weight001/21/{\sqrt{2}}111117/5\sqrt{17}/53/53/5114411

Hence by Simpson’s rule

I=∫0π21-1625⁢sin2⁡θ⁢d⁢θ≈π12⁢(1+45⁢17+35)≈1.28.I=\int_{0}^{\frac{\pi}{2}}\sqrt{1-\frac{16}{25}\sin^{2}\theta}\,d\theta\approx% {\frac{\pi}{12}\left(1+\frac{4}{5}\sqrt{17}+\frac{3}{5}\right)\approx 1.28.}

Thus the perimeter of the ellipse is 20⁢I≈25.6⁢m.20I\approx{25.6m.}