Find the exact area of the surface obtained by rotating the curve about the x-axis. y = x³, 0 ≤ x ≤ 2

Solution:

Given, y = x³, 0 ≤ x ≤ 2

We have to find the area of the surface by rotating the curve about the x-axis.

For rotation about the x-axis, the surface area formula is given by

\(S=2\pi \int_{a}^{b}y\sqrt{1+(y')^{2}}\, dx\)

y = x³

y’ = 3x²

By rotating the curve y = x³ about the x-axis in the interval [0, 2]

\(S=2\pi \int_{0}^{2}(x^{3})\sqrt{1+(3x^{2})^{2}}\, dx\)

\(S=2\pi \int_{0}^{2}(x^{3})\sqrt{1+9x^{4}}\, dx\)

Let u = 1 + 9x⁴

du = 36x³ dx

dx = du/36x³

Substituting u and du in the integral,

\(S=2\pi \int_{0}^{2}(x^{3})\sqrt{u}\frac{du}{36x^{3}}\)

\(S=\frac{2\pi }{36} \int_{0}^{2}\sqrt{u}\, du\)

\(S=\frac{2\pi }{36}\left [ \frac{2u^{\frac{3}{2}}}{3} \right ]_{0}^{2}\)

\(S=\frac{2\pi }{36}\frac{2}{3}\left [ u\frac{3}{2} \right ]_{0}^{2}\)

\(S=\frac{\pi }{27}[(1+9x^{4})^{\frac{3}{2}}]_{0}^{2}\)

\(S=\frac{\pi }{27}[(1+9(2)^{4})^{\frac{3}{2}}-(1+9(0)^{4})^{\frac{3}{2}}]\)

\(S=\frac{\pi }{27}[(145)^{\frac{3}{2}}-(1)^{\frac{3}{2}}]\)

\(S=\frac{\pi }{27}[1746.03-1]\)

\(S=\frac{\pi }{27}[1745.03]\)

\(S=64.67\pi\)

\(S=64.67(3.14)\)

S = 203.06 square units.

Therefore, the exact area of the surface is 203.06 square units.

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Find the exact area of the surface obtained by rotating the curve about the x-axis. y = x³, 0 ≤ x ≤ 2

Summary:

The exact area of the surface obtained by rotating the curve about the x-axis. y = x³, 0 ≤ x ≤ 2 is 203.06 square units.