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Dr. Geek also owns some champagne flutes, which are shaped exactly like the shape obtained by revolving the graph of y=lnxxfrom x=0.6to x=5around the x-axis, as shown in the figure and measured in inches. Given that the volume of the shape obtained from revolving faround the x-axis on[a,b]can be calculated with the formula πabfx2dxabout how much champagne can each glass hold?

Short Answer

Expert verified

Each glass can hold 1.582in3champagne.

Step by step solution

01

Step 1. Given information

y=lnxx fromx=0.6tox=5.

02

Step 2. The given formula is, src="data:image/svg+xml;base64,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" role="math" localid="1650636453545" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/b0e6a9b7-5277-4ad1-aa47-649de4918102.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220423%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220423T061642Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=c6dddfee4916ed7e460f972f5417fb6661fac7c9e6c65fa708915951b118c74b" π∫abfx2dx.

Substitute the known values in the formula.

πabfx2dx=πablnxx2dx.

To solve the integral, let us use the integrate by parts method.

Integrating by parts, fg'=fg-f'g.

Here, f=ln2x,g'=1x2.

role="math" localid="1650637610294" f'=2lnxx,g=-1xπablnxx2dx=π-ln2xx-2-2lnxx2dx=π-ln2xx+2-lnxx--1x2dx=π-ln2xx-2lnxx+21x2dx=π-ln2xx-2lnxx-2x+C=-πln2x+2lnx+2x+C

03

Step 3. Let us use the limit values to find the definite integral.

π0.65fx2dx=0.65lnxx2dx=-πln2x+2lnx+2x0.65=-πln20.6+2ln0.6+20.6--πln25+2ln5+25=1.582in3

Therefore, the required value is,1.582in3.

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