Chapter 5: Q. 86 (page 430)
Dr. Geek also owns some champagne flutes, which are shaped exactly like the shape obtained by revolving the graph of from to around the -axis, as shown in the figure and measured in inches. Given that the volume of the shape obtained from revolving around the -axis oncan be calculated with the formula about how much champagne can each glass hold?
Short Answer
Each glass can hold champagne.
Step by step solution
Step 1. Given information
fromto.
Step 2. The given formula is, 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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.
.
To solve the integral, let us use the integrate by parts method.
Integrating by parts, .
Here, .
role="math" localid="1650637610294"
Step 3. Let us use the limit values to find the definite integral.
Therefore, the required value is,.
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