All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.


Graph of 5i
Graph of 5i

Since x=asin\theta, we have dx=acos\theta ~d\theta
When x= \frac{\sqrt{3}a}{2}, \theta = \frac{\pi}{3}
When x= \frac{a}{2}, \theta = \frac{\pi}{6}

\int_{\pi / 6}^{\pi / 3} \sqrt{1-sin^{2}\theta}~(acos\theta) ~d\theta

= a \int_{\pi / 6}^{\pi / 3} cos^{2} \theta ~d \theta

= a \int_{\pi / 6}^{\pi / 3} \frac{1+cos2\theta}{2} ~d \theta

= \frac{a}{2}[\theta + \frac{sin 2 \theta}{2}]\biggl|_{\pi / 6}^{\pi / 3}

= \frac{a \pi}{12}

KS Comments:

The function is a periodic curve and some students do have problem reading such functions. They should be alert of the domain that they are required to draw too. Some students were unsure how the hint that f(x+3a)=f(x) should help. The parametric integration should be quite simple to handle as students just need to be cautious and introduce the double angle formula from MF15 effectively.


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