Find the volume of the region formed by y=x^2-7x+10 and y=x+3 rotated about the y-axis. I was able to graph it, but I'm having difficulty when trying to come up with the integral for it.

prkosnognm

prkosnognm

Answered question

2022-07-21

Finding the volume of a region rotated about the y-axis.
I'm having trouble trying to find the volume of the region formed by y = x 2 7 x + 10 and y = x + 3 rotated about the y-axis. I was able to graph it, but I'm having difficulty when trying to come up with the integral for it.
Any help showing me how to get the integral I should use would be a huge help.

Answer & Explanation

Tolamaes04

Tolamaes04

Beginner2022-07-22Added 12 answers

Step 1
This one may be easier to handle using the Method of Cylindrical Shells. Setting x + 3 = x 2 7 x + 10 we find that the curves meet at x = 1 and x = 7. Over the interval [1,7], the line y = x + 3 is the upper curve. When we take a vertical strip of width "dx" from x to x + d x, and rotate it, we get a cylindrical shell of radius x and height ( x + 3 ) ( x 2 7 x + 10 ) = x 2 + 8 x 7.
Step 2
Thus the volume is 1 7 2 π x ( x 2 + 8 x 7 ) d x .
For the integration, multiply out and integrate term by term.
Jaylene Hunter

Jaylene Hunter

Beginner2022-07-23Added 2 answers

Explanation:
You must split the integration. for y [ 9 4 , 4 ] the radii of the inner and outer annuli are the two roots of x 2 7 x + ( 10 y ) = 0, whereas for y [ 4 , 10 ] the outer radius is the larger root of that equation, but the inner radius is given by y 3

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