The diagram below shows the region $R$ bounded by the circle with equation ${{\left( x-4 \right)}^{2}}+{{y}^{2}}=9$, the line with equation $y=-\sqrt{5}x+3\sqrt{5}$ and the $x$-axis. The line and the circle meets at the point $A$ with coordinates $\left( 2,\sqrt{5} \right)$.
<figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1760348786349-85pfzds54e6.png" width="400" class="aligncenter" /></figure>
Show that the volume $V$, formed when $R$ is rotated through $2\pi $ radians about the $y$-axis is given by
$\pi \int_{0}^{\sqrt{5}}{\left[ \frac{{{\left( y-3\sqrt{5} \right)}^{2}}}{5}-\left( 25-{{y}^{2}}-8\sqrt{9-{{y}^{2}}} \right) \right]}\mathrm{ d}y$.
Hence, evaluate $V$, leaving your answer correct to $3$ decimal places.[5]

The diagram below shows the region $R$ bounded by the circle with equation ${{\left( x-4 \right)}^{2}}+{{y}^{2}}=9$, the line with equation $y=-\sqrt{5}x+3\sqrt{5}$ and the $x$-axis. The line and the circle meets at the point $A$ with coordinates $\left( 2,\sqrt{5} \right)$.

Show that the volume $V$, formed when $R$ is rotated through $2\pi $ radians about the $y$-axis is given by

$\pi \displaystyle\displaystyle \int_{0}^{\sqrt{5}}{\left[ \frac{{{\left( y-3\sqrt{5} \right)}^{2}}}{5}-\left( 25-{{y}^{2}}-8\sqrt{9-{{y}^{2}}} \right) \right]}\mathrm{ d}y$.

Hence, evaluate $V$, leaving your answer correct to $3$ decimal places.

Applications of Intergration - General

Applications of Intergration

H2 Math

International Baccalaureate Higher Level Mathematics

2023 SAJC P1 Q5

2023 SAJC P1 Q5

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