A region bounded by a logarithm, a line…

Name: ______________________ Date: ____________ A region bounded by a logarithm, a line and the x-axis (9 points)

Graph

Let

R

be the region in the first quadrant bounded by the graph of

y = \ln x

, the line

y = 4 - x

and the

x

-axis, as shown. The graph of

y = \ln x

and the line meet at the point

(c, \ln c)

. A graphing calculator is required.

(a) Find the area of $R$ . [2 points]
(b) Find the volume of the solid generated when $R$ is revolved about the vertical line $x = 5$ . [3 points]
(c) Region $R$ is the base of a solid. For this solid, each cross section perpendicular to the $x$ -axis is a semicircle whose diameter lies in $R$ . Find the volume of the solid. [2 points]
(d) The horizontal line $y = k$ divides $R$ into two regions of equal area. Write, but do not solve, an equation involving one or more integrals whose solution gives $k$ . [2 points]

The question