anonymous
  • anonymous
Find the volume of the solid generated by revolving the region bounded by the x-axis and the curve xsinx , about 0
Mathematics
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anonymous
  • anonymous
Find the volume of the solid generated by revolving the region bounded by the x-axis and the curve xsinx , about 0
Mathematics
schrodinger
  • schrodinger
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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dumbcow
  • dumbcow
For parts a) and c) i will use the disc method because to use shell method when revolving around horizontal axis means getting function in terms of y. y = xsin(x) --> x = f(y) = ? see what i mean part a) |dw:1328793504021:dw| each circular cross-section is a ring with outer radius of 3 inner radius of 3-f(x) = 3-xsinx \[V =\pi \int\limits_{0}^{\pi}R^{2} -r^{2} \] \[=\pi \int\limits_{0}^{\pi}3^{2}-(3-x \sin x)^{2} dx\] \[=\pi \int\limits_{0}^{\pi}6x \sin x-x^{2}\sin^{2} x dx\] part b) |dw:1328794143959:dw| the radius is distance from line of revolution as x goes from 0 to pi, radius goes from 2pi to pi --> r = 2pi-x height is just the y_value or f(x) \[V = 2\pi \int\limits_{0}^{\pi}(2\pi-x)(x \sin x) dx\] part c) |dw:1328794750169:dw| each circular cross-section is a ring with outer radius of 2+f(x) = 2+xsinx inner radius of 2 \[V =\pi \int\limits_{0}^{\pi}R^{2} -r^{2} \] \[=\pi \int\limits_{0}^{\pi}(2+x sin x)^{2}-2^{2} dx\] \[=\pi \int\limits_{0}^{\pi}4x \sin x+x^{2}\sin^{2} x dx\] part d) |dw:1328795102992:dw| the radius is distance from line of revolution (x=-2pi) as x goes from 0 to pi, radius goes from 2pi to 3pi --> r = 2pi+x height is just the y_value or f(x) \[V = 2\pi \int\limits_{0}^{\pi}(2\pi+x)(x \sin x) dx\]

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