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Rechner für den Flächeninhalt einer Rotationsfläche

Berechnen Sie den Oberflächeninhalt einer Rotationsfläche Schritt für Schritt

Der Rechner berechnet den Oberflächeninhalt der Rotationsfläche (um die angegebene Achse) einer expliziten, polaren oder parametrischen Kurve auf dem angegebenen Intervall und zeigt die Rechenschritte an.

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.