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Знайди об'єм тіла, отриманого при обертанні навколо осі абсцис криволінійної трапеції, обмеженої лініями: y=x^(2)+12,x=0,x=4,y=0

Знайди об'єм тіла, отриманого при обертанні навколо осі абсцис криволінійної трапеції, обмеженої лініями:\newliney=x2+12,x=0,x=4,y=0 y=x^{2}+12, x=0, x=4, y=0

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Q. Знайди об'єм тіла, отриманого при обертанні навколо осі абсцис криволінійної трапеції, обмеженої лініями:\newliney=x2+12,x=0,x=4,y=0 y=x^{2}+12, x=0, x=4, y=0
  1. Identify Boundaries: Identify the boundaries of the region to be rotated. The region is bounded by y=x2+12y = x^2 + 12, x=0x = 0, x=4x = 4, and y=0y = 0. The rotation is around the xx-axis.
  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[y(x)]2 dx\pi[y(x)]^2 \, dx from the lower to the upper x-boundary.\newlineV=π∫x=0x=4(x2+12)2 dxV = \pi \int_{x=0}^{x=4} (x^2 + 12)^2 \, dx.
  3. Expand and Simplify: Expand the integrand and simplify the integral. \newline(x2+12)2=x4+24x2+144(x^2 + 12)^2 = x^4 + 24x^2 + 144.\newlineSo, V=π∫x=0x=4(x4+24x2+144) dxV = \pi \int_{x=0}^{x=4} (x^4 + 24x^2 + 144) \, dx.
  4. Compute Integral: Compute the integral.\newline∫04x4 dx=x55∣04=10245\int_0^4 x^4 \, dx = \left.\frac{x^5}{5}\right|_0^4 = \frac{1024}{5},\newline∫0424x2 dx=24x33∣04=7683\int_0^4 24x^2 \, dx = \left.24\frac{x^3}{3}\right|_0^4 = \frac{768}{3},\newline∫04144 dx=144x∣04=576\int_0^4 144 \, dx = 144x \bigg|_0^4 = 576.\newlineV=π(10245+7683+576)V = \pi\left(\frac{1024}{5} + \frac{768}{3} + 576\right).
  5. Calculate Final Volume: Calculate the final volume.\newlineV=π(204.8+256+576)=π(1036.8)V = \pi(204.8 + 256 + 576) = \pi(1036.8).

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