The container is made by rotating y=1.5x^{2} where 0 leq x leq 3 about the axis x=-0.5 (the bottom is flat).

daniellex0x0xto

daniellex0x0xto

Answered question

2022-08-11

Finding volume of solid revolution about x-axis
I'm asked to find the volume of a container of height 13.5 The container is made by rotating y = 1.5 x 2 where 0 x 3 about the axis x = 0.5 (the bottom is flat).
I tried finding the right volume a number of ways, but I believe my most solid attempt so far is this:
I invert the function to get y = 2 3 x .
From this I tried to calculate π 0 3 2 3 x d x which gave me 3 π which I thought was correct, but it wasnt. Plotting this, I realized the height becomes 3 when I integrate the inverted function from 0 to 3, so I thought I should maybe integrate from 0 to 13.5. In that case I get 243 π 4 which is also incorrect. I'm sure I've misinterpreted what it means to rotate the function about the axis x = 0.5

Answer & Explanation

kunstboom8w

kunstboom8w

Beginner2022-08-12Added 8 answers

Explanation:
I translated 0.5 units to the right so I had the equation y = 3 2 ( x 1 2 ) 2 to rotate around y axis and I got V = π 0 3 ( 3 2 ( x 1 2 ) 2 ) 2 d x = 14067 π 320 138.1

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