Integral Volume Calculator With Steps, Volume Of A Solid Of Revolution

Volume of a Solid of Revolution

How to find the volume of a solid of revolution generated by revolving a region bounded by the graph of a function around one of the axes using definite integrals? We will present examples based on the methods of disks and washers where the integration is parallel to the axis of rotation. A set of exercises with answers is presented at the end.

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Formulas to calculate the volume generated by revolving graphs of functions around one of the axes

Formula 1 – Disk around the x axis

If f is a function such that f(x) ≥ 0 for all x in the interval , the volume of the solid generated by revolving, around the x axis, the region bounded by the graph of f, the x axis (y = 0) and the vertical lines x = x1 and x = 2 is given by the integral

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Figure 9. Torus generated when the circle with center at (0,R) and radius r is rotated around the x axis

Solution to Example 5

The equation of the circle is given byx 2 + (y – R) 2 = r 2Solve the above equation for y to obtain two solutions each for one semicircley = R + √(r 2 – x 2) , upper semi circle , and y = R – √(r 2 – x 2) , lower semi circleThe torus is generated by rotating the two halves semi circles the x axis hence the use of formula 2 given above to find the volume of the torus. Let f(x) = R + √(r 2 – x 2) and h(x) = R – √(r 2 – x 2). Because of the symmetry of the circle and therefore the torus with respect to the y axis, we integrate from x = 0 to x = r then double the answer to find the total volume.

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ext{Volume} = int_{0}^{r} pi < f(x)^2 - h(x)^2 > dx \\= pi int_{0}^{r} < (R + sqrt(r^2 - x^2))^2 - (R - sqrt(r^2 - x^2))^2 >dx \\= 4 R pi int_{0}^{r} < sqrt(r^2 - x^2) >dx = 4 R pi (1/2) left < x sqrt{r^2-x^2} + r^2arctan(frac{x}{sqrt{r^2-x^2}}) ight >_0^r\\= 4 R pi ( pi r^2 / 4) = {pi}^2 R r^2

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Exercises

(1) Find the volume of the solid generated when the region between the graphs of f(x) = x 2 + 2 and h(x) = x is revolved about the x axis and over the interval <0,1>.(2) Find the volume generated when the finite region bounded by the curves y = x3, y = x2 is revolved about the y axis.(hint: you need to find the points of intersections of the two curves)

Answers to Above Exercises

(1) 26 Pi /5(2) pi /10

More Links and References

Volume by Cylindrical Shells Method.integrals and their applications in calculus.Area under a curve.Area between two curves.

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