contestada

Using disks or washers, find the volume of the solid obtained by rotating the region bounded by the curves y=1/x, y=0, x=1, and x=4 about the line y=−1.

Respuesta :

erturkmemmedli

Answer:

[tex]\frac{3}{4}\pi +4\pi\ln2[/tex]

Step-by-step explanation:

[tex]y = \frac{1}{x} ,\:\:y=0,\:\:x=1,\:\:x=4[/tex]

about the line y = -1

[tex]V=\pi\int\limits^4_1[(\frac{1}{x}+1)^2-(0+1)^2]\:dx=\pi\int\limits^4_1(\frac{1}{x^2}+\frac{2}{x}+1-1)\:dx=\\\\=\pi(-\frac{1}{x}+2\ln x)|^4_1=\pi(-\frac{1}{4} +2\ln 4+1-2\ln 1)=\frac{3}{4}\pi +4\pi\ln2[/tex]

Cricetus

The volume of the solid will be "[tex]\frac{3}{4} \pi +4 \pi \ ln \ 2[/tex]".

Given curves:

  • [tex]y = \frac{1}{x}[/tex]
  • [tex]y =0[/tex]
  • [tex]x =1[/tex]
  • [tex]x = 4[/tex]

About the line "y = -1".

The volume will be:

→ [tex]V = \pi \int\limits^4_1 [(\frac{1}{x}+1 )^2-(0-1)^2]dx[/tex]

      [tex]=\pi \int\limits^4_1 (\frac{1}{x^2}+\frac{2}{x} +1-1 )dx[/tex]

      [tex]= \pi (-\frac{1}{x}+2 ln x )|_1^4[/tex]

      [tex]= \pi (- \frac{1}{4} +2 ln4+1-2 ln 1 )[/tex]

      [tex]= \frac{3}{4} \pi +4 \pi \ ln \ 2[/tex]

Thus the above response is right.  

Learn more about volume here:

https://brainly.com/question/9291616

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