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

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)

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