How do you find the volume of the solid obtained by rotating the region bounded by the curves about the x-axis?

Question

#y=2x, y=x^2#

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Answer 1 · 2018-07-02T20:48:50

#\frac{64\pi}{15}\ \text{unit}^3#

The given curves: #y=2x# & #y=x^2# intersect each other at two points #(0,0)# & #(2, 4)#

Now, the volume of the solid obtained by revolving the region bounded by straight line #y_1=2x# & the upward parabola #y_2=x^2#

#=\int_0^2 \pi(y_1^2-y_2^2)\ dx #

#=\int_0^2 \pi((2x)^2-(x^2)^2)\ dx #

#=\pi\int_0^2 (4x^2-x^4)\ dx #

#=\pi(4/3x^3-{x^5}/5)_0^2 #

#=\pi(32/3-32/5)#

#=\frac{64\pi}{15}\ \text{unit}^3#