Mathematics | Area of the surface of solid of revolution

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:

  1. Cartesian Form:

    • Area of solid formed by revolving the arc of curve about x-axis is-
      S= \int_{x=a}^{x=b} 2\pi y\sqrt{1+(\frac{dy}{dx})^2}dx
    • Area of revolution by revolving the curve about y axis is-
      S= \int_{y=c}^{y=d} 2\pi x \sqrt{1+(\frac{dx}{dy})^2}dy

  2. Parametric Form: x=x(t), y=y(t)

    • About x-axis:
      S=\int_{t=t_{1}}^{t=t_{2}} 2\pi y(t) \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}dt
    • About y-axis:
      S=\int_{t=t_{1}}^{t=t_{2}} 2\pi x(t) \sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}dt

  3. Polar Form: r=f(θ)

    • About the x-axis: initial line  \theta = \frac{\pi}{2}
      S= \int_{\theta=\theta_1}^{\theta _2}2\pi y\frac{ds}{d\theta}d\theta
      =\int_{\theta=\theta_1}^{\theta _2}2\pi (r sin\theta) \sqrt{r^2+(\frac {dr}{d\theta})^2}d\theta
      Here replace r by f(θ)
    • About the y-axis:
      S= \int_{\theta=\theta_1}^{\theta _2}2\pi x\frac{ds}{d\theta}d\theta
      =\int_{\theta=\theta_1}^{\theta _2}2\pi (r cos\theta) \sqrt{r^2+(\frac {dr}{d\theta})^2}d\theta
      Here replace r by f(θ)

  4. About any axis or line L: S= \int 2\pi (PM) ds 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
      S=\int_{x=a}^{x=b} 2\pi (PM)\sqrt{1+(\frac{dy}{dx})^2}dx
      Here PM is in terms of x.
    • Limits for y: y = c to y = d
      S= \int_{y=c}^{y=d} 2\pi (PM)\sqrt{1+(\frac{dx}{dy})^2}dy
      Here PM is in terms of y.
  5. Example:
    Find the area of the solid of revolution generated by revolving the parabola y^2=4ax, 0\leq x \leq 3a 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:

    S= \int_{x=a}^{x=b} 2\pi y\sqrt{1+(\frac{dy}{dx})^2}dx

    Here y^2= 4ax. Now we need to calculate dy/dx

    Differentiating w.r.t x we get:

    2yy'= 4a


    1+(y')^2=1+\frac {4a^2}{y^2}=\frac{y^2+4a^2}{y^2}

    Using y^2=4ax

    \sqrt {1+(y')^2}=\sqrt{\frac{4ax+4a^2}{y^2}}=\frac{2\sqrt a}{y}\sqrt {a+x}

    Now we are provided with limits of x as x=0 to x=3. Plugging our calculated values in the above formula we get:

    S=\int_{0}^{3a} 2\pi y.{\frac{2\sqrt a}{y}\sqrt {a+x}}dx

    =2\pi\int_{0}^{3a} y.{\frac{2\sqrt a}{y}\sqrt {a+x}}dx

    =4\pi\sqrt a\int_{0}^{3a}\sqrt {a+x}

    =4\pi\sqrt a\int_{0}^{3a}\frac{2}{3}(x+a)^{3/2}\Biggr|_{0}^{3a}

    =\frac{8}{3}\pi\sqrt a ((4a)^{3/2}-(a)^{3/2})

    =\frac{8}{3}\pi\sqrt a.a^{3/2}(8-1)

    =\frac{56\pi a^2}{3} sq. units

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