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  3. 電卓: 微分積分学 II
  4. 微積分計算機

回転面の面積計算機

回転面の表面積をステップバイステップで計算する

この計算機は、与えられた区間における明示形、極形式、または媒介変数表示の曲線を、指定された軸のまわりに回転して得られる回転曲面の表面積を、手順を示しながら求めます。

Choose type:

Enter a function:

Rotate around the -axis

Enter a lower limit:

If you need `-oo`, type -inf.

Enter an upper limit:

If you need `oo`, type inf.

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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.


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