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Kalkulator Luas Permukaan Hasil Putar
Hitung luas permukaan benda putar langkah demi langkah
Kalkulator akan menghitung luas permukaan putar (mengelilingi sumbu yang diberikan) dari kurva eksplisit, polar, atau parametrik pada interval yang diberikan, dengan langkah-langkah ditunjukkan.
Solution
Your input: find the area of the surface of revolution of $$$f\left(x\right)=x^{2}$$$ rotated about the x-axis on $$$\left[0,1\right]$$$
The surface area of the curve is given by $$$S = 2\pi \int_a^b f \left(x\right) \sqrt{\left(f'\left(x\right)\right)^2+1}d x$$$
First, find the derivative: $$$f '\left(x\right)=\left(x^{2}\right)'=2 x$$$ (steps can be seen here)
Finally, calculate the integral $$$S = \int_{0}^{1} 2 \pi x^{2} \sqrt{\left(2 x\right)^{2} + 1} d x=\int_{0}^{1} 2 \pi x^{2} \sqrt{4 x^{2} + 1} d x$$$
The calculations and the answer for the integral can be seen here.