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# The region R is bounded by the x-axis, x = 1, x = 3, and y=√(x-1).

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The region R is bounded by the x-axis, x = 1, x = 3, and y=√(x-1).

a. Find the area of R.

b. Find the average value of the region.

c. Find the volume of the solid generated when R is revolved about the x-axis.

Apr 20, 2020

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a)
$$Area = \int_a^bf(x)dx\\ \int_1^3\sqrt{x-1}dx=\frac{4\sqrt{2}}{3}$$

b)

$$Average\;Value = \frac{1}{b-a}\int_a^bf(x)dx\\ \frac{1}{3-1}\int_1^3\sqrt{x-1}dx=\frac{1}{2}*\frac{4\sqrt{2}}{3}=\frac{2\sqrt{2}}{3}$$

c)

$$Volume = \int_a^bA(x)dx=\int_a^b\pi r(x)^2dx=\pi\int_a^br(x)^2dx\\ \pi\int_1^3(\sqrt{x-1})^2dx=2\pi$$

Apr 20, 2020