By using the substitution $x=3\sec \theta $, evaluate \(\displaystyle\displaystyle \displaystyle\i

Question

By using the substitution $x=3\sec \theta $, evaluate $\displaystyle\displaystyle \displaystyle\int_{3\sqrt{2}}^{6}{\frac{3x+1}{\sqrt{{{x}^{2}}-9}}\mathrm{d}x}$ exactly.[5] The diagram shows an ellipse with equation $\frac{{{x}^{2}}}{16}+\frac{{{\left( y-2 \right)}^{2}}}{4}=1$. Find the area of the shaded region, giving your answer correct to $3$ decimal places.[2] Find the exact volume of the solid generated when the shaded region is rotated $180{}^\circ $ about the $y$-axis.[4]

Accepted Answer

(a) $9\sqrt{3} - 9 + \ln\frac{2+\sqrt{3}}{\sqrt{2}+1}$ (b)(i) Area = $10.753$ units$^2$ (ii) Volume = $\frac{64}{3}\pi - \frac{7\sqrt{7}}{3}\pi$ units$^3$

2017 YJC P1 Q9

2017 YJC P1 Q9

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