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SMK Teknologi dan Rekayasa

Soal 130 Volume Benda Putar


Jika\ daerah\ yang\ diarsir\ pada\ gambar\ di\ bawah\ ini\ diputar\\mengelilingi\ sumbu\ X\ sejauh\ 360^{\circ},\ maka\ volume\ benda\ putar\\yang\ terjadi\ adalah\ ....

a.\ \ 168\pi \ satuan\ volume\ \ \ \ \ \ \ \ \ \ \ b.\ \ 166\pi \ satuan\ volume\ \ \ \ \ \ \ \ \ \ \ c.\ \ 164\pi \ satuan\ volume\\d.\ \ 162\pi \ satuan\ volume\ \ \ \ \ \ e.\ \ 160\pi \ satuan\ volume
Jawaban: a

x-y+4=0\ \ \ \ \ \ \to \ \ \ \ \ \ y=x+4
Karena\ pemutaran\ mengelilingi\ sumbu\ X,\ maka:
Volume\ benda\ putar=\pi \int_{a}^{b}y^{2}\ dx=\pi \int_{-2}^{4}(x+4)^{2}\ dx
(Dalam\ perhitungannya\ penulis\ tunjukkan\ teknik\ substitusi)

\begin{array}{rcl}Misalkan\ u=x+4&\to &du=(1+0)\ dx=dx\\Untuk\ x=-2&\to &u=-2+4=2\\Untuk\ x=4&\to &u=4+4=8\end{array}

\begin{array}{rcl}Volume\ benda\ putar&=&\pi \int_{2}^{8}u^{2}\ du\\\\&=&\pi \left [ \frac{u^{3}}{3} \right ]_{2}^{8}\\\\&=&\frac{1}{3}\pi \left [ u^{3} \right ]_{2}^{8}\\\\&=&\frac{1}{3}\pi \left [ 8^{3}-2^{3} \right ]\\\\&=&\frac{1}{3}\pi \left [ 512-8 \right ]\\\\&=&\frac{1}{3}\pi \times 504\\\\&=&168\pi \end{array}

Volume\ benda\ putar\ yang\ terjadi\ adalah\ 168\pi \ satuan\ volume

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