Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.


d. Write an integral for the volume of the solid.

Verified step by step guidance

1. Step 1: Understand the problem. The region R is bounded by the graphs of y = 1 + √x, x = 4, and y = 1. The goal is to find the volume of the solid formed when this region is revolved about the y-axis using the washer method.
2. Step 2: Express x in terms of y for the curve y = 1 + √x. Rearrange the equation to isolate √x: √x = y - 1. Then square both sides to get x = (y - 1)^2.
3. Step 3: Determine the bounds for y. The region starts at y = 1 (the horizontal line) and ends at y = 1 + √4 = 3 (since x = 4). Thus, the bounds for y are from 1 to 3.
4. Step 4: Set up the washer method formula for the volume. The volume of the solid is given by the integral: V = π ∫[y₁ to y₂] [(Outer radius)^2 - (Inner radius)^2] dy. Here, the outer radius is x = (y - 1)^2, and the inner radius is x = 4.
5. Step 5: Write the integral. Substitute the bounds and radii into the formula: V = π ∫[1 to 3] [(y - 1)^4 - 4^2] dy. This integral represents the volume of the solid formed by revolving the region R about the y-axis.

Key Concepts

Solid of Revolution

A solid of revolution is a three-dimensional shape created by rotating a two-dimensional area around an axis. In this case, the region R is revolved around the y-axis, which means the volume of the solid can be calculated using methods such as the disk or washer method, depending on the shape of the cross-sections.

Washer Method

The washer method is a technique used to find the volume of a solid of revolution when the cross-sections are washers (disks with holes). It involves integrating the area of the outer radius minus the area of the inner radius, typically expressed as π(R^2 - r^2), where R and r are the outer and inner radii, respectively.

Definite Integral

A definite integral is a mathematical expression that calculates the accumulation of quantities, such as area or volume, over a specific interval. In this context, the definite integral will be set up to evaluate the volume of the solid formed by revolving region R around the y-axis, integrating from the lower to the upper bounds of the region.