Question

Find the volume of the solid obtained by rotating the region in the first quadrant bounded by the given curve

about the y-axis.
y = 0.25 - (x-4)²

Answer 1

4π/3 ≈ 4.18879 cubic units

Explanation:

You want the volume of revolution formed by revolving the 1st-quadrant area under y = 0.25 -(x -4)² around the y-axis.

Volume

There are several ways to find the volume of revolution. One can integrate a differential of volume, where that differential is a cylindrical shell or a washer. Alternatively, one can multiply the area under the curve by the centerline length of the path of revolution. We choose this latter approach.

Area

The area in the first quadrant of the region being rotated can be found as the integral between its x-intercepts.

The x-intercepts are found by ...

0 = 1/4 -(x -4)²

x -4 = ±√(1/4) = ±1/2

x = 4 ± 1/2 = {3.5, 4.5} . . . . . . x-intercepts

Then the area is ...

`\displaystyle\int_(3.5)^(4.5)(0.25-(x-4)^2)\,dx=\left[0.25x-(1)/(3)(x-4)^3\right]_(3.5)^(4.5)\\\\\\=0.25(1)-(1)/(3)(0.5^3-(-0.5)^3)=(1)/(6)\qquad\text{square units}`

Revolved

The radius from the y-axis to the line of symmetry of the area is 4 units, so the length of the rotation path is

C = 2πr = 2π(4) = 8π

The product of this path length and the area is ...

V = C·A = (8π)(1/6) = 4π/3

The volume of the solid is 4π/3 cubic units.

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Additional comment

The area under a symmetrical portion of a parabola is 2/3 of the area of the enclosing rectangle. The portion of the parabola in the 1st quadrant is 1 unit wide and 0.25 units high, so its area is (1)(1/4)(2/3) = 1/6 square unit, as we found above.

The second attachment shows the volume found using the cylindrical shell method.

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