Finding surface area of revolution of a parametric curve around a vertical axis
The formula for surface area of revolution of a parametric curve
The surface area of the solid created by revolving a parametric curve around the ???y???-axis is given by
???S_x=\int^b_a 2\pi{x}\sqrt{\left[f'(t)\right]^2+\left[g'(t)\right]^2}\ dt???
where the curve is defined over the interval ???[a,b]???,
where ???f'(t)??? is the derivative of the curve ???x=f(t)???
where ???g'(t)??? is the derivative of the curve ???y=g(t)???
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How to find surface area of revolution with a parametric curve around a vertical axis of revolution
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Finding surface area of the parametric curve rotated around the y-axis
Example
Find the surface area of revolution of the solid created when the parametric curve is rotated around the given axis over the given interval.
???x=2t^2???
???y=2t^3???
for ???0\le{t}\le3???, rotated around the ???y???-axis
We’ll call the parametric equations
???f(t)=2t^2???
???g(t)=2t^3???
The limits of integration are defined in the problem, but we need to find both derivatives before we can plug into the formula.
???f'(t)=4t???
???g'(t)=6t^2???
Now we’ll plug into the formula for the surface area of revolution.
???S_y=\int^3_02\pi\left(2t^2\right)\sqrt{\left(4t\right)^2+\left(6t^2\right)^2}\ dt???
???S_y=\int^3_04\pi{t^2}\sqrt{16t^2+36t^4}\ dt???
???S_y=\int^3_04\pi{t^2}\sqrt{4t^2\left(4+9t^2\right)}\ dt???
???S_y=\int^3_0 8\pi{t^3}\sqrt{4+9t^2}\ dt???
???S_y=8\pi\int^3_0t^3\sqrt{4+9t^2}\ dt???
We use different formulas to find the surface area of revolution of a parametric curve, depending on whether the axis of revolution is horizontal or vertical.
We’ll use u-substitution, letting
???u=4+9t^2???
???9t^2=u-4???
???t^2=\frac{u-4}{9}???
???du=18t\ dt???
???dt=\frac{du}{18t}???
We’ll make the substitution.
???S_y=8\pi\int^{t=3}_{t=0}t^3\sqrt{u}\ \frac{du}{18t}???
???S_y=\frac{8\pi}{18}\int^{t=3}_{t=0}t^2\sqrt{u}\ du???
???S_y=\frac{8\pi}{18}\int^{t=3}_{t=0}\frac{u-4}{9}\sqrt{u}\ du???
???S_y=\frac{8\pi}{18}\int^{t=3}_{t=0}\left(\frac{u}{9}-\frac49\right)u^{\frac12}\ du???
???S_y=\frac{8\pi}{18}\int^{t=3}_{t=0}\frac{u^{\frac32}}{9}-\frac{4u^{\frac12}}{9}\ du???
???S_y=\frac{8\pi}{162}\int^{t=3}_{t=0}u^{\frac32}-4u^{\frac12}\ du???
???S_y=\frac{4\pi}{81}\left(\frac25u^{\frac52}-\frac83u^{\frac32}\right)\bigg|^{t=3}_{t=0}???
Back-substituting for ???u???, we get
???S_y=\frac{4\pi}{81}\left[\frac25\left(4+9t^2\right)^{\frac52}-\frac83\left(4+9t^2\right)^{\frac32}\right]\bigg|^3_0???
Evaluate over the interval.
???S_y=\frac{4\pi}{81}\left[\frac25\left(4+9(3)^2\right)^{\frac52}-\frac83\left(4+9(3)^2\right)^{\frac32}\right]-\frac{4\pi}{81}\left[\frac25\left(4+9(0)^2\right)^{\frac52}-\frac83\left(4+9(0)^2\right)^{\frac32}\right]???
???S_y=\frac{4\pi}{81}\left[\frac25\left(4+81\right)^{\frac52}-\frac83\left(4+81\right)^{\frac32}\right]-\frac{4\pi}{81}\left[\frac25\left(4+0\right)^{\frac52}-\frac83\left(4+0\right)^{\frac32}\right]???
???S_y=\frac{4\pi}{81}\left[\frac25\left(85\right)^{\frac52}-\frac83\left(85\right)^{\frac32}-\frac25\left(4\right)^{\frac52}+\frac83\left(4\right)^{\frac32}\right]???
???S_y=\frac{4\pi}{81}\left[\frac25\left[\left(85\right)^5\right]^\frac12-\frac83\left[\left(85\right)^3\right]^\frac12-\frac25\left[\left(4\right)^{\frac12}\right]^5+\frac83\left[\left(4\right)^{\frac12}\right]^3\right]???
???S_y=\frac{4\pi}{81}\left[\frac25\left[85\left(85\right)^4\right]^\frac12-\frac83\left[85\left(85\right)^2\right]^\frac12-\frac25(2)^5+\frac83(2)^3\right]???
???S_y=\frac{4\pi}{81}\left[\frac25\left[\left(85\right)^2\sqrt{85}\right]-\frac83\left[85\sqrt{85}\right]-\frac25(32)+\frac83(8)\right]???
???S_y=\frac{4\pi}{81}\left[\frac{2\left(85\right)^2\sqrt{85}}{5}-\frac{680\sqrt{85}}{3}-\frac{64}{5}+\frac{64}{3}\right]???
???S_y=\frac{4\pi}{81}\left[\frac{2\cdot5\cdot5\cdot17\cdot17\cdot\sqrt{85}}{5}-\frac{680\sqrt{85}}{3}-\frac{64}{5}+\frac{64}{3}\right]???
???S_y=\frac{4\pi}{81}\left[2,890\sqrt{85}-\frac{680\sqrt{85}}{3}-\frac{64}{5}+\frac{64}{3}\right]???
???S_y=\frac{4\pi}{81}\left[2,890\sqrt{85}-\frac{64}{5}+\frac{64-680\sqrt{85}}{3}\right]???
Find a common denominator.
???S_y=\frac{4\pi}{81}\left[\frac{43,350\sqrt{85}}{15}-\frac{192}{15}+\frac{320-3,400\sqrt{85}}{15}\right]???
???S_y=\frac{4\pi}{81}\left(\frac{43,350\sqrt{85}-3,400\sqrt{85}-192+320}{15}\right)???
???S_y=\frac{4\pi}{81}\left(\frac{39,950\sqrt{85}+128}{15}\right)???
