contestada

Consider the region satisfying the inequalities.y ≤ e−x, y ≥ 0, x ≥ 0a) Find area of regionb) Find the volume of the solid generated by revolving the region about the x-axis.c) Find the volume of the solid generated by revolving the region about the y-axis.

Respuesta :

LammettHash
  • Revolving about the [tex]x[/tex]-axis:

Using the disk method, the volume is

[tex]\displaystyle\pi\int_0^\infty e^{-2x}\,\mathrm dx=\boxed{\frac\pi2}[/tex]

Alternatively, using the shell method, the volume is

[tex]\displaystyle2\pi\int_0^1y(-\ln y)\,\mathrm dy=\frac\pi2[/tex]

  • Revolving about the [tex]y[/tex]-axis:

Using the shell method, the volume is

[tex]\displaystyle2\pi\int_0^\infty xe^{-x}\,\mathrm dx=\boxed{2\pi}[/tex]

Alternatively, using the disk method, the volume is

[tex]\displaystyle\pi\int_0^1(-\ln x)^2\,\mathrm dx=2\pi[/tex]

Otras preguntas