Calculadora de área de superficie de revolución
Calcular el área de una superficie de revolución paso a paso
La calculadora encontrará el área de la superficie de revolución (alrededor del eje dado) de la curva explícita, polar o paramétrica en el intervalo dado, con pasos mostrados.
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.