Question

Let R be the region bounded by the graphs of y = ln(x) and y = 4x − 6. (a) Find the area of R. (Round your answer to three decimal places.) (b) Find the volume of the solid generated when R is rotated about the horizontal line y = −6. (Round your answer to four decimal places.) (c) Write, but do not evaluate, an expression involving one or more integrals that can be used to find the volume of the solid generated when R is revolved about the y-axis. (Round your answer for the upper limit of the integral to three decimal

Answer 1

(a) The area of region R bounded by the graphs of y = ln(x) and y = 4x - 6 is approximately 4.371 square units.

(b) The volume of the solid generated when R is rotated about the horizontal line y = -6is approximately 241.511 cubic units.

(c) An expression involving one or more integrals to find the volume of the solid generated when R is revolved about the y-axis is
`\(V = \pi \int_(a)^(b) [f(x)]^2 - [g(x)]^2 \,dx\)`, wheref(x) and g(x) are the upper and lower functions, respectively, and \(a\) and \(b\) are the limits of integration.

Step-by-step explanation:

(a) To find the area of region R between y = ln(x) and y = 4x - 6, we need to set the two functions equal to each other and find the x-coordinate of their intersection. Solve 4x - 6 = ln(x) to find (x), and then integrate the difference of the two functions over the interval [a, b], where a and b are the x-coordinates of the intersection points.

(b) To find the volume of the solid generated by revolving R about y = -6, we can use the disk method. Set up the integral
`\(V = \pi \int_(a)^(b) [f(x) + 6]^2 \,dx - \pi \int_(a)^(b) [g(x) + 6]^2 \,dx\)`, where f(x) and g(x) are the upper and lower functions, respectively. Evaluate this integral over the interval [a, b] found in part (a).

(c) To express the volume of the solid when R is revolved about the y-axis, we use the washer method. The expression is
`\(V = \pi \int_(a)^(b) [f(x)]^2 - [g(x)]^2 \,dx\)`, where f(x) and g(x) are the radial distances to the outer and inner curves, respectively. The limits of integration (a) and (b) are determined by the x-values where the curves intersect.