Find the volume generated by rotating the region bounded by y = x, x = 1, and y2 = 4x, about the x-axis.
Given, the region is bounded by y=x,y2=4xy=x, y^2=4xy=x,y2=4x and x=1x=1x=1 .
Volume generated is given by,
v=π∫ab([f(x)]2−[g(x)]2)dx=π∫01((2x)2−x2)dx=π∫01(4x−x2)=π(2x2−x33)01=π(2−13)=5π3v=\pi\int_a^b ([f(x)]^ 2−[g(x)] ^2)dx\\ =\pi\int_0^1((2\sqrt x)^ 2−x^ 2 )dx\\ =\pi\int_0^1(4x−x ^2)\\ =\pi(2x ^2− \frac{x^3}{3})_0^1 \\ =\pi(2− \frac{1}{3})\\ = \frac{5\pi}{3} v=π∫ab([f(x)]2−[g(x)]2)dx=π∫01((2x)2−x2)dx=π∫01(4x−x2)=π(2x2−3x3)01=π(2−31)=35π
Thus, volume is 5π3\frac{5\pi}{3}35π .
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