Final answer:
The volumes of the solids formed by revolving a region around the x-axis and the line y=1 are calculated using integral calculus. For the x-axis, a cylindrical volume is determined by integrating the square of x, and for the line y=1, the volume is the difference between two cylindrical volumes, considering the hole created by revolving around y=1.
Step-by-step explanation:
To find the volume of a solid generated by revolving a region around an axis, we use methods from integral calculus, specifically, the disk or washer method. The student is asked to find the volumes of solids generated by revolving the region in the first quadrant bounded by the curve x and the y-axis about two different axes: the x-axis and the line y=1.
- Volume when revolved around the x-axis: This will form a cylinder. Since there are no specific bounds given for x, we assume that there is some upper limit to the region. If we call this limit b, then the volume V is calculated by integrating the area of the circular cross-section perpendicular to the x-axis from 0 to b. Using the formula V = π∫[0 to b] x2 dx, we can find the volume.
- Volume when revolved around the line y=1: This revolves the area into a washer-shaped solid. The volume is the outer solid minus the inner solid. The outer solid is the same cylinder as before, but the inner solid is a cylinder with a hole in it. The volume of the hollow part corresponds to revolving the region between y=0 and y=1 around the line y=1, which can be calculated by the same method but taking into account the distance from the line y=1.
To answer CHECK YOUR UNDERSTANDING 1.5, the volume of a sphere of radius r is 4πr3/3.