Using the substitution $u=\sqrt{4x+1}$, show that\(\displaystyle\displaystyle \displaystyle\int{\f

Question

Using the substitution $u=\sqrt{4x+1}$, show that$\displaystyle\displaystyle \displaystyle\int{\frac{\sqrt{4x+1}}{x-2}}\mathrm{ d}x=\displaystyle\displaystyle \displaystyle\int{\frac{2{{u}^{2}}}{{{u}^{2}}-9}}\mathrm{ d}u$.[3] The region $R$ lies in the first quadrant and is bounded by the curve $y=\frac{\sqrt{4x+1}}{x-2}$ and the lines $x=6$, $x=12$ and $y=0.5$. Sketch the graph of $y=\frac{\sqrt{4x+1}}{x-2}$ for $x \geqslant 0$, giving the coordinates of any intercepts with the axes and the equations of any asymptotes. Shade the region $R$ on your sketch. [3] Find the exact area of $R$, giving your answer in the form $a+b\ln c$.[5] Find the volume of the solid generated when $R$ is rotated through $2\pi $ radians about the $x$- axis. Give your answer correct to $2$ decimal places. [3]

Accepted Answer

(a) Shown; (b)(i) Graph with VA x=2, HA y=0; (b)(ii) 1 + 3ln(8/5) units²; (b)(iii) 11.04 units³

N2024 P1 Q10

N2024 P1 Q10

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