Find the area of the surface generated by revolving the parabola y^2=2ax

Answer: The area of the surface generated by revolving the parabola y²=2ax about x-axis bounded by x=a is equal to 2πa²(3√3-1)/3 unit.

In this page, we study the surface area generated by revolving a curve (parabola) about x-axis with formula. The formulas for the volume are given below:

Formula

• The formula for the area A of the surface generated by revolving a curve y=f(x) about x-axis bounded by x=a is: 

A = 2π $\displaystyle \int_0^a f(x) \sqrt{1+\left(\dfrac{dy}{dx} \right)^2}dx$.

•  The formula for the area A of the surface generated by revolving a curve x=f(y) about y-axis bounded by y=b is: 

A = 2π$\displaystyle \int_0^b f(y)\sqrt{1+\left(\dfrac{dx}{dy} \right)^2}dy$.

Question

Find the surface area generated by revolving the parabola y²=2ax about x-axis bounded by x=a.

Answer:

Lets express the parabola y²=2ax in the form y=f(x).

y²=2ax

⇒ $y= \sqrt{2ax}$

Thus, we have f(x) = $\sqrt{2ax}$

Now, differentiating both sides of y²=2ax with respect to x, we have:

$2y \dfrac{dy}{dx}=2a$ ⇒ $\dfrac{dy}{dx}=\dfrac{a}{y}$ ⇒ $\dfrac{dy}{dx}=\dfrac{a}{\sqrt{2ax}}$, since $y= \sqrt{2ax}$.

Thus, using the above formula for the area A generated by revolving the parabola y²=2ax about x-axis bounded by x=a is given by

Area A = 2π $\displaystyle \int_0^a f(x) \sqrt{1+\left(\dfrac{dy}{dx} \right)^2}dx$

= 2π $\displaystyle \int_0^a \sqrt{2ax} \sqrt{1+\dfrac{a^2}{2ax}} dx$

= 2π $\displaystyle \int_0^a \sqrt{2ax+a^2} dx$

= 2π $\left[ \dfrac{(2ax+a^2)^{\frac{3}{2}}}{\frac{3}{2} \cdot 2a}\right]_0^a$

= 2π $ \cdot \dfrac{2}{3} \cdot \dfrac{1}{2a}$ $\left[ (2a \cdot a+a^2)^{\frac{3}{2}} -(0+a^2)^{\frac{3}{2}} \right]$

= $\dfrac{2\pi}{3a} \left\{ (3a^2)^{\frac{3}{2}} -(a^2)^{\frac{3}{2}} \right\}$

= $\dfrac{2\pi}{3a} \left\{ 3\sqrt{3}a^3 -a^3 \right\}$

= $\dfrac{2\pi a^2}{3} (3\sqrt{3}-1)$ unit.

Therefore, the area of the surface generated by revolving the parabola y²=2ax about x-axis is $\dfrac{2\pi a^2}{3} (3\sqrt{3}-1)$ unit.

Related Article: Find the volume generated by revolving the parabola y²=2ax about y-axis

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