Solid of Rotation:I Need Verification

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In summary, to determine the volume of a solid obtained by rotating the region bounded by y=x^2-2x and y=x about the line y=4, we can use the formula v = pi int(0, 3)[ (y2 - 4)^2 - (y1 - 4)^2 ] dx, where y1 = x and y2 = x^2 - 2x. By solving for the intersection points of these two functions, we find that the straight line is closer to y=4 than the parabola between x=0 and x=3. After evaluating the integral, we get a volume of approximately 30.6 pi.
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zcabral
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Homework Statement


Determine the volume of a sold obtained by rotating the region bounded by y=x^2-2x and y=x about the line y =4

Homework Equations


The Attempt at a Solution


Let:
y1 = x
y2 = x^2 - 2x.
v be the volume.

y1 and y2 meet where:
x^2 - 2x = x
x^2 - 3x = 0
x(x - 3) = 0
x = 0, x = 3.

The straight line is closer to y = 4 than the parabola between x = 0 and x = 3.

v = pi int(0, 3)[ (y2 - 4)^2 - (y1 - 4)^2 ] dx
= pi int(0, 3) [ (x^2 - 2x - 4)^2 - (x - 4)^2 ] dx
= pi int(0, 3) [ x^4 + 4x^2 + 16 - 4x^3 + 16x - 8x^2 - x^2 - 16 + 8x ] dx
= pi int(0, 3) [ x^4 - 4x^3 - 5x^2 + 24x ] dx
= pi [ x^5 / 5 - x^4 - 5x^3 / 3 + 12x^2 ](0, 3)
= pi [ 3^5 / 5 - 3^4 - 5 * 3^3 / 3 + 12 * 3^2 ]
= 30.6 pi.

IS THIS RIGHT?
 
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  • #2
I got the same answer.
 

What is a solid of rotation?

A solid of rotation is a three-dimensional shape created by rotating a two-dimensional shape around an axis. The resulting shape is symmetrical and has a circular cross-section.

How is the volume of a solid of rotation calculated?

The volume of a solid of rotation can be calculated using the formula V = π∫ab (f(x))2 dx, where a and b are the limits of rotation and f(x) is the equation of the curve being rotated.

What are some examples of solids of rotation?

Some common examples of solids of rotation include cylinders, cones, and spheres. Other examples include tori, wine glasses, and vases.

What is the difference between a solid of rotation and a solid of revolution?

A solid of rotation is created by rotating a two-dimensional shape around an axis, while a solid of revolution is created by rotating a two-dimensional shape around a line. Both result in symmetrical three-dimensional shapes, but the axis or line of rotation is different.

What is the significance of solids of rotation in real life?

Solids of rotation have many practical applications in engineering and design. For example, they are used in the construction of water towers, bottles, and pipes. They also play a role in physics and calculus, as they are used to calculate the volume of irregular shapes and to model rotational motion.

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