Jawab:
Penjelasan dengan langkah-langkah:
- Mencari titik singgung dan jarak antar titik:
√x = 3x => x = 1/9 => y = 1/3 => P1 = (1/9, 1/3)
y² = 3y-10 => y = 2 => x = 4 => P3 = (4,2)
3x = (10-x)/3 => x = 1, y = 3 => P2 = (1,3)
[tex]a = \dfrac{8}{9}\cdot\sqrt{10}, b = |P_3-P_2|= \sqrt{(4-1)^2+(2-3)^2}\\ b = \sqrt{3^2+1^2} = \sqrt{10}[/tex]
- Mencari garis yang melalui P1 dan P3 :
[tex]\displaystyle y - 2 = \frac{2-\frac{1}{3}}{4-\frac{1}{9}} \cdot(x-4)\\\\y = \frac{3}{7} \cdot(x-4)+2 = \dfrac{3x+2}{7}[/tex]
- Mencari Luas nya :
[tex]\displaystyle L = \dfrac{a\cdot b}{2} - \int\limits_{\tfrac{1}{9}}^4 \left(\sqrt{x}-\frac{3x+2}{7}\right)\; dx\\L = 5\cdot 1\cdot \dfrac{8}{9} - \left( \frac{2}{3}\left(\sqrt{4^3} - \dfrac{1}{\sqrt{9^3}} \right) - \dfrac{3}{14}\left(4^2 - \dfrac{1}{9^2}\right)-\dfrac{2}{7} \left(4-\frac{1}{9} \right) \right)\\[/tex]
[tex]\Huge{\boxed{\boxed{\boldsymbol{L = \frac{595}{162}\mathrm{\; satuan\; unit} }}}}[/tex]
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