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Знайди об'єм тіла, отриманого при обертанні навколо осїабсцис фігури, обмеженої лініями: y=6x^(2),y=6x Відповідь: V= ◻ pi

Знайди об'єм тіла, отриманого при обертанні навколо осїабсцис фігури, обмеженої лініями:\newliney=6x2,y=6x y=6 x^{2}, y=6 x \newlineВідповідь: V= V= □ \square π \pi

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Q. Знайди об'єм тіла, отриманого при обертанні навколо осїабсцис фігури, обмеженої лініями:\newliney=6x2,y=6x y=6 x^{2}, y=6 x \newlineВідповідь: V= V= □ \square π \pi
  1. Identify Intersection Points: Identify the bounds of integration by setting the equations equal to each other to find the points of intersection.\newline6x2=6x6x^2 = 6x\newlinex2=xx^2 = x\newlinex(x−1)=0x(x - 1) = 0\newlinex=0,x=1x = 0, x = 1
  2. Set Up Integral for Volume: Set up the integral for the volume using the method of washers. The outer radius RR is given by y=6xy = 6x and the inner radius rr by y=6x2y = 6x^2.V=π∫x=0x=1(R2−r2) dxV = \pi \int_{x=0}^{x=1} (R^2 - r^2) \, dxV=π∫x=0x=1((6x)2−(6x2)2) dxV = \pi \int_{x=0}^{x=1} ((6x)^2 - (6x^2)^2) \, dx
  3. Simplify and Integrate: Simplify the integrand and perform the integration.\newlineV=π∫x=0x=1(36x2−36x4) dxV = \pi \int_{x=0}^{x=1} (36x^2 - 36x^4) \, dx\newlineV=π[12x3−(36/5)x5]01V = \pi [12x^3 - (36/5)x^5]_{0}^{1}\newlineV=π[12(1)3−(36/5)(1)5]−π[12(0)3−(36/5)(0)5]V = \pi [12(1)^3 - (36/5)(1)^5] - \pi [12(0)^3 - (36/5)(0)^5]\newlineV=π[12−36/5]V = \pi [12 - 36/5]\newlineV=π[60/5−36/5]V = \pi [60/5 - 36/5]\newlineV=π[24/5]V = \pi [24/5]

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