Mencari Rumus Luas Permukaan Bangun Torus dengan Integral

Setelah sebelumnya kita membahas rumus volume Torus, pada kesempatan kali ini saya lanjutkan untuk mencari rumus luas permukaan bangun tersebut. Untuk rumus volume torus bisa dilihat pada postingan sebelumnya (klik di sini).

Secara geometris bangun torus seperti gambar berikut ini

Terdapat dua jari-jari pada torus yaitu jari-jari besar (R) dan jari-jari kecil (r).

LUAS PERMUKAAN TORUS

Untuk mencari luas permukaan bangun Torus kita akan menggunakan rumus luas permukaan benda putar. untuk bangun torus, gambarnya sebagai berikut:

Lingkaran dengan diameter $\left( {R - r} \right)$ dengan titik pusat $\left( {0,\frac{1}{2}\left( {R + r} \right)} \right)$ memotong sumbu $Y$ di titik $\left( {0,R} \right)$ dan $\left( {0,r} \right)$ diputar sebesar ${360^o}$ terhadap sumbu $X$. Pesamaan lingkaran tersebut adalah sebagai berikut:

$\begin{array}{l}{\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} = {r^2}\\ \Leftrightarrow {\left( {x - 0} \right)^2} + {\left( {y - \frac{1}{2}\left( {R + r} \right)} \right)^2} = {\left( {\frac{1}{2}\left( {R - r} \right)} \right)^2}\\ \Leftrightarrow {x^2} + {\left( {y - \frac{1}{2}\left( {R + r} \right)} \right)^2} = {\left( {\frac{1}{2}\left( {R - r} \right)} \right)^2}\\ \Leftrightarrow {\left( {y - \frac{1}{2}\left( {R + r} \right)} \right)^2} = {\left( {\frac{1}{2}\left( {R - r} \right)} \right)^2} - {x^2}\\ \Leftrightarrow y - \frac{1}{2}\left( {R + r} \right) = \sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}} \\ \Leftrightarrow y = \frac{1}{2}\left( {R + r} \right) \pm \sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}} \\ \Leftrightarrow {y_1} = \frac{1}{2}\left( {R + r} \right) + \sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}  \vee {y_2} = \frac{1}{2}\left( {R + r} \right) - \sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}} \end{array}$

Terdapat 2 persamaan bola yaitu yang berwarna merah untuk ${y_1}$ dan biru untuk ${y_2}$. Dari persamaan lingkaran tersebut, kita akan mencari rumus luas permukaan dengan dua langkah. Pertama mencari luas permukaan bagian luar torus dan kedua mencari luas permukaan bagian dalam torus. Kedua hasil perhitungan kemudian dijumlahkan menghasilkan luas permukaan total untuk Torus.

LUAS PERMUKAAN LUAR

$\begin{array}{l}{L_1} = 2\pi \int\limits_a^b {f\left( x \right)\sqrt {1 + {{\left[ {f'\left( x \right)} \right]}^2}} } dx\\{L_1} = 2\pi \int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {{y_1}\sqrt {1 + {{\left( {{y_1}'} \right)}^2}} } dx\\{L_1} = 2\pi \int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {\left( {\frac{1}{2}\left( {R + r} \right) + \sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}} } \right)\sqrt {1 + {{\left( {\frac{{ - x}}{{\sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}} }}} \right)}^2}} } dx\\{L_1} = 2\pi \int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {\left( {\frac{1}{2}\left( {R + r} \right) + \sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}} } \right)\sqrt {1 + \frac{{{x^2}}}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}} } dx\\{L_1} = 2\pi \int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {\left( {\frac{1}{2}\left( {R + r} \right)\sqrt {1 + \frac{{{x^2}}}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}}  + \sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}} .\sqrt {1 + \frac{{{x^2}}}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}} } \right)} dx\\{L_1} = 2\pi \int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {\left( {\frac{1}{2}\left( {R + r} \right)\sqrt {\frac{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2}}}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}}  + \sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}.\frac{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2}}}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}} } \right)} dx\\{L_1} = 2\pi \int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {\left( {\frac{1}{2}\left( {R + r} \right).\frac{1}{2}\left( {R - r} \right)\sqrt {\frac{1}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}}  + \sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2}} } \right)} dx\\{L_1} = 2\pi \int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {\left( {\frac{1}{4}\left( {{R^2} - {r^2}} \right)\sqrt {\frac{1}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}}  + \frac{1}{2}\left( {R - r} \right)} \right)} dx\\{L_1} = \frac{1}{2}\pi \left( {{R^2} - {r^2}} \right)\int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {\sqrt {\frac{1}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}} } dx + \pi \left( {R - r} \right)\int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {dx} \\{L_1} = \frac{1}{2}\pi \left( {{R^2} - {r^2}} \right)\left[ {{{\sin }^{ - 1}}\left( {\frac{x}{{\frac{1}{2}\left( {R - r} \right)}}} \right)} \right]_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} + \pi \left( {R - r} \right)\left[ x \right]_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)}\\{L_1} = \frac{1}{2}\pi \left( {{R^2} - {r^2}} \right)\left[ {{{\sin }^{ - 1}}\left( {\frac{{\frac{1}{2}\left( {R - r} \right)}}{{\frac{1}{2}\left( {R - r} \right)}}} \right) - {{\sin }^{ - 1}}\left( {\frac{{ - \frac{1}{2}\left( {R - r} \right)}}{{\frac{1}{2}\left( {R - r} \right)}}} \right)} \right] + \pi \left( {R - r} \right)\left[ {\frac{1}{2}\left( {R - r} \right) - \left( {\frac{1}{2}\left( {R - r} \right)} \right)} \right]\\{L_1} = \frac{1}{2}\pi \left( {{R^2} - {r^2}} \right)\left[ {{{\sin }^{ - 1}}\left( 1 \right) - {{\sin }^{ - 1}}\left( { - 1} \right)} \right] + \pi \left( {R - r} \right)\left( {R - r} \right)\\{L_1} = \frac{1}{2}\pi \left( {{R^2} - {r^2}} \right)\left[ {\frac{\pi }{2} - \left( { - \frac{\pi }{2}} \right)} \right] + \pi {\left( {R - r} \right)^2}\\{L_1} = \frac{1}{2}{\pi ^2}\left( {{R^2} - {r^2}} \right) + \pi {\left( {R - r} \right)^2}\end{array}$

LUAS PERMUKAAN DALAM

$\begin{array}{l}{L_2} = 2\pi \int\limits_a^b {f\left( x \right)\sqrt {1 + {{\left[ {f'\left( x \right)} \right]}^2}} } dx\\{L_2} = 2\pi \int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {{y_2}\sqrt {1 + {{\left( {{y_2}'} \right)}^2}} } dx\\{L_2} = 2\pi \int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {\left( {\frac{1}{2}\left( {R + r} \right) - \sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}} } \right)\sqrt {1 + {{\left( {\frac{x}{{\sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}} }}} \right)}^2}} } dx\\{L_2} = 2\pi \int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {\left( {\frac{1}{2}\left( {R + r} \right) - \sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}} } \right)\sqrt {1 + \frac{{{x^2}}}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}} } dx\\{L_2} = 2\pi \int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {\left( {\frac{1}{2}\left( {R + r} \right)\sqrt {1 + \frac{{{x^2}}}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}}  - \sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}} .\sqrt {1 + \frac{{{x^2}}}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}} } \right)} dx\\{L_2} = 2\pi \int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {\left( {\frac{1}{2}\left( {R + r} \right)\sqrt {\frac{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2}}}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}}  - \sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}.\frac{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2}}}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}} } \right)} dx\\{L_2} = 2\pi \int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {\left( {\frac{1}{2}\left( {R + r} \right).\frac{1}{2}\left( {R - r} \right)\sqrt {\frac{1}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}}  - \sqrt {{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2}} } \right)} dx\\{L_2} = 2\pi \int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {\left( {\frac{1}{4}\left( {{R^2} - {r^2}} \right)\sqrt {\frac{1}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}}  - \frac{1}{2}\left( {R - r} \right)} \right)} dx\\{L_2} = \frac{1}{2}\pi \left( {{R^2} - {r^2}} \right)\int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {\sqrt {\frac{1}{{{{\left( {\frac{1}{2}\left( {R - r} \right)} \right)}^2} - {x^2}}}} } dx - \pi \left( {R - r} \right)\int\limits_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} {dx} \\{L_2} = \frac{1}{2}\pi \left( {{R^2} - {r^2}} \right)\left[ {{{\sin }^{ - 1}}\left( {\frac{x}{{\frac{1}{2}\left( {R - r} \right)}}} \right)} \right]_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)} - \pi \left( {R - r} \right)\left[ x \right]_{ - \frac{1}{2}\left( {R - r} \right)}^{\frac{1}{2}\left( {R - r} \right)}\\{L_2} = \frac{1}{2}\pi \left( {{R^2} - {r^2}} \right)\left[ {{{\sin }^{ - 1}}\left( {\frac{{\frac{1}{2}\left( {R - r} \right)}}{{\frac{1}{2}\left( {R - r} \right)}}} \right) - {{\sin }^{ - 1}}\left( {\frac{{ - \frac{1}{2}\left( {R - r} \right)}}{{\frac{1}{2}\left( {R - r} \right)}}} \right)} \right] - \pi \left( {R - r} \right)\left[ {\frac{1}{2}\left( {R - r} \right) - \left( {\frac{1}{2}\left( {R - r} \right)} \right)} \right]\\{L_2} = \frac{1}{2}\pi \left( {{R^2} - {r^2}} \right)\left[ {{{\sin }^{ - 1}}\left( 1 \right) - {{\sin }^{ - 1}}\left( { - 1} \right)} \right] - \pi \left( {R - r} \right)\left( {R - r} \right)\\{L_2} = \frac{1}{2}\pi \left( {{R^2} - {r^2}} \right)\left[ {\frac{\pi }{2} - \left( { - \frac{\pi }{2}} \right)} \right] - \pi {\left( {R - r} \right)^2}\\{L_2} = \frac{1}{2}{\pi ^2}\left( {{R^2} - {r^2}} \right) - \pi {\left( {R - r} \right)^2}\end{array}$

LUAS PERMUKAAN TOTAL

$\begin{array}{l}L = {L_1} + {L_2}\\L = \frac{1}{2}{\pi ^2}\left( {{R^2} - {r^2}} \right) + \pi {\left( {R - r} \right)^2} + \frac{1}{2}{\pi ^2}\left( {{R^2} - {r^2}} \right) - \pi {\left( {R - r} \right)^2}\\L = {\pi ^2}\left( {{R^2} - {r^2}} \right)\end{array}$

KESIMPULAN

Luas permukaan bangun Torus yang memiliki jari-jari luar (R) dan jari-Jari dalam (r) adalah

\[L = {\pi ^2}\left( {{R^2} - {r^2}} \right)\]

Demikian pembahasannya dan terima kasih.