Answer to Question #205262 in Calculus for carl

Question #205262

Find the volume generated by rotating the region bounded by y = x, x = 1, and y2 = 4x, about the x-axis.

1
Expert's answer
2021-06-12T05:09:46-0400

Given, the region is bounded by y=x,y2=4xy=x, y^2=4x and x=1x=1 .

Volume generated is given by,

v=πab([f(x)]2[g(x)]2)dx=π01((2x)2x2)dx=π01(4xx2)=π(2x2x33)01=π(213)=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} ​

Thus, volume is 5π3\frac{5\pi}{3} .


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