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Determine which of the following integrals represents the volume of the solid by rotating the region enclosed by?

1) ∫(0 to 2π) ∫(0 to 3) r dr dθ
2) ∫(0 to 2π) ∫(0 to 3) r² dr dθ
3) ∫(0 to 2π) ∫(0 to 3) r³ dr dθ
4) ∫(0 to 2π) ∫(0 to 3) r⁴ dr dθ

1 Answer

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Final answer:

The integral that represents the volume of the solid obtained by rotating the region enclosed by the given equations is option 3) ∫(0 to 2π) ∫(0 to 3) r³ dr dθ.

Step-by-step explanation:

The integral that represents the volume of the solid obtained by rotating the region enclosed by the given equations is option 3) ∫(0 to 2π) ∫(0 to 3) r³ dr dθ.



To determine the integral that represents the volume of the solid, we need to use the formula for the volume of a solid of revolution. The formula is V = ∫(a to b) A(x) dx, where A(x) is the area of the cross-section of the solid at a given x-value. In this case, the cross-sections are circular and have a radius of r, so the area of each cross-section is A(x) = πr².



Substituting A(x) = πr² into the formula for the volume, we get V = ∫(a to b) πr² dx. In polar coordinates, the element of area is given by dA = r dr dθ, so we can rewrite the volume formula as V = ∫(0 to 2π) ∫(0 to R) r² dr dθ, where R is the maximum value of r. To obtain the volume of the solid, we need to evaluate this double integral.

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