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Calculadora de Área da Superfície de Revolução

Calcular a área de uma superfície de revolução passo a passo

A calculadora encontrará a área da superfície de revolução (em torno do eixo dado) da curva explícita, polar ou paramétrica no intervalo dado, com as etapas mostradas.

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.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

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.