Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines.

(i) x-axis; (ii) the line

Question

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines.

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(i) x-axis; (ii) the line

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BlackTom AI · Accepted Answer

The region is bounded by y = x^2 and y = 20x − x^2. These two parabolas meet where x^2 = 20x − x^2 ⇒ 2x^2 − 20x = 0 ⇒ x( x − 10 ) = 0, so the region lies between x = 0 and x = 10. For 0 < x < 10 the upper curve is y = 20x − x^2 and the lower curve is y = x^2.

(i) Revolve about the x-axis.
- Relevant concept: washer method (disks with holes) about a horizontal axis. For a fixed x the outer radius is R(x) = y_upper = 20x − x^2, and the inner radius is r(x) = y_lower = x^2.
- Volume: V = π ∫_{0}^{10} [R(x)^2 − r(x)^2] dx
  = π ∫_0^{10} [ (20x − x^2)^2 − (x^2)^2 ] dx.
- Expand and simplify integrand:
  (20x − x^2)^2 = 400x^2 − 40x^3 + x^4, subtract x^4 gives 400x^2 − 40x^3.
- Integrate:
  ∫_0^{10} (400x^2 − 40x^3) dx = 400·(x^3/3)|_0^{10} − 40·(x^4/4)|_0^{10}
  = (400/3)·1000 − 10·10000 = 400000/3 − 100000 = 100000/3.
- Multiply by π: V = (100000/3)π.

(ii) Revolve about the vertical line x = 10.
- Relevant concept: cylindrical shells about a vertical axis. Use vertical strips at x (0 ≤ x ≤ 10). Each strip of height h(x) = y_upper − y_lower = (20x − x^2) − x^2 = 20x − 2x^2 = 2x(10 − x). The shell radius is 10 − x, and shell circumference is 2π(10 − x).
- Volume by shells: V = ∫_{0}^{10} 2π(radius)(height) dx = 2π ∫_0^{10} (10 − x)(20x − 2x^2) dx.
  Factor: 20x − 2x^2 = 2x(10 − x), so integrand = (10 − x)·2x(10 − x) = 2x(10 − x)^2.
  Thus V = 2π ∫_0^{10} 2x(10 − x)^2 dx = 4π ∫_0^{10} x(10 − x)^2 dx.
- Expand and integrate:
  x(10 − x)^2 = x(100 − 20x + x^2) = 100x − 20x^2 + x^3.
  ∫_0^{10} (100x − 20x^2 + x^3) dx = [50x^2 − (20/3)x^3 + (1/4)x^4]_0^{10}
  = 50·100 − (20/3)·1000 + (1/4)·10000 = 5000 − 20000/3 + 2500 = 2500 − 20000/3 = 2500/3.
- Multiply by 4π: V = 4π · (2500/3) = (10000/3)π.

Why these are correct and why other choices are incorrect:
- Any answer that fails to square the functions correctly in the washer method for (i) or that uses washers instead of shells incorrectly for the vertical-axis rotation in (ii) will give wrong coefficients. The algebra above yields exact rational multiples of π; choices with different numerators/denominators (e.g. 104000/3 π, or omitting the factor from the shell integrand) are therefore incorrect.

Final answers:
(i) V = (100000/3)π
(ii) V = (10000/3)π

These values follow directly from the step-by-step integrals above.

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. (i) x-axis; (ii) the line Single choice

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Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. ​ ​ ​ (i) x-axis; (ii) the line ​Single choice