Find the volume of the solid obtained by rotating the region bounded by? Calculus?

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Find the volume of the solid obtained by rotating the region bounded by y=18x-3x^2 and y=0 about the y axis.

I know we use the cylinder method but I keep getting the wrong answe

✅ Answers

? Best Answer

  • y = 18x − 3x² = 3x (6 − x)

    Curve intersects line y = 0 at x = 0 and x = 6

    V = 2π ∫₀⁶ x * (18x − 3x²) dx
    V = 2π ∫₀⁶ (18x² − 3x³) dx
    V = 2π (6x³ − 3x⁴/4) |₀⁶
    V = 2π * ((1296 − 972) − (0 − 0))
    V = 648π

  • Using the cylindrical shell method, volume = definite integral (2*pi*radius*height)dx for values of x from a to b, where a and b are the x-coordinates of intersection with the x-axis and a is greater than or equal to 0. Therefore, volume = definite integral (2*pi*x*(18x-3x^2))dx for values of x from 0 to 6. Make sure to have a greater than or equal to 0, not equal to a negative number.

    Source(s):
    B.C. Calc class

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