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Volumes Of Solid Revolutions

1)Find the volume when the areas bounded by the curves y=sin(x), x=0, x=π, y=0 rotated about x-axis.

Sol:

 V=\(\prod \int_{0}^{\prod }y^{2}dx \)

V= \(\int_{0}^{\prod }(sin(x))^{2}dx\)

V= \(\int_{0}^{\prod }\frac{(1-cos(2x))}{2}dx\)

V=π/2 unit³


2) Find the volume when the areas bounded by the curves y=√x , x=4 y=0 and Revolved about y-axis.

Solutions: 

y=√x and x=4 intersect at a point (4, 2) and y^2=x.

V=\(\prod \int_{0}^{2 }\left ( 4^{2}-x^{2} \right )dy\)

V=\(\prod \int_{0}^{2 }\left ( 16 - \left ( y^{2} \right )^{2} \right )dy\)

V=\(\prod \int_{0}^{2 }\left ( 16 - y^{4} \right )dy\)

V=\( \frac{128* \prod}{5} \)unit³

3)Find the volume when the areas bounded by the curves y=x^4 and y=x . Rotated about y-axis.

Sol:

y=\(x^{4}\) and y=x intersect at (0, 0) and (1,1)

y=\(x^{4}\) then \(x_{1}=y^{\frac{1}{4}}\)

y=x  then x2=y

V=\(\prod \int_{0}^{1} (x_{1}^{2}-x_{2}^{2})dy\)

V=\(\prod \int_{0}^{1}(√y-y²)dy\)

V=π/3

 

4) Find the Volume of the regions bounded by the curves y=x², y=2x rotated about line x=2.

Sol:

y=x² and y=2x intersect at (0, 0) and (2, 4) .

y=2x then x1=y/2

y=x² then x2 = √y

V=\( \prod \int_{0}^{4} ((2-x_{1})² -(2-x_{2})²)dy\)

now you can simplify further.



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