Consider a plane y=f(x) in the x-y plane between ordinates x=a and x=b. If a certain portion of this curve is revolved about an axis, a solid of revolution is generated.
We can calculate the area of this revolution in various ways such as:
Area of solid formed by revolving the arc of curve about x-axis is-
Area of revolution by revolving the curve about y axis is-
- Area of solid formed by revolving the arc of curve about x-axis is-
- About x-axis:
Polar Form: r=f(θ)
About the x-axis: initial line
Here replace r by f(θ)
About the y-axis:
Here replace r by f(θ)
- About the x-axis: initial line
About any axis or line L: where PM is the perpendicular distance of a point P of the curve to the given axis.
Limits for x: x = a to x = b
Here PM is in terms of x.
Limits for y: y = c to y = d
Here PM is in terms of y.
- Limits for x: x = a to x = b
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Find the area of the solid of revolution generated by revolving the parabola about the x-axis.
Now we are given with the Cartesian form of the equation of parabola and the parabola has been rotated about the x-axis. Hence we use the formula for revolving Cartesian form about x-axis which is:
Here . Now we need to calculate dy/dx
Differentiating w.r.t x we get:
Now we are provided with limits of x as x=0 to x=3. Plugging our calculated values in the above formula we get:
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