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

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

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Q. Обчисли об'єм тіла, отриманого при обертанні навколо осі абсцис фігури, обмеженої лініями:\newliney=6x2,y=6x y=6 x^{2}, y=6 x \newlineВідповідь: V=□π V=\square \pi
  1. Identify Bounds and Region: Identify the bounds of integration and the region to be rotated.\newlineThe curves intersect where 6x2=6x6x^2 = 6x. Solving for xx, we get x2=xx^2 = x, so x(x−1)=0x(x-1) = 0. Thus, x=0x = 0 or x=1x = 1. The region is between x=0x = 0 and x=1x = 1.
  2. Set up Integral: Set up the integral for the volume using the disk method.\newlineThe volume VV of the solid of revolution is given by the integral of π(R2−r2) dx\pi(R^2 - r^2) \, dx from aa to bb, where RR is the outer radius and rr is the inner radius. Here, R=6xR = 6x and r=6x2r = 6x^2.
  3. Calculate Integral: Calculate the integral.\newlineV=π∫01((6x)2−(6x2)2)dx=π∫01(36x2−36x4)dx.V = \pi\int_{0}^{1} ((6x)^2 - (6x^2)^2) dx = \pi\int_{0}^{1} (36x^2 - 36x^4) dx.
  4. Simplify and Integrate: Simplify and integrate. V=π∫01(36x2−36x4)dx=π[12x3−7.2x5]V = \pi\int_{0}^{1} (36x^2 - 36x^4) dx = \pi[12x^3 - 7.2x^5] from 00 to 11.

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