Final answer:
To calculate the volume of a solid formed by rotating a region around the y-axis in the first quadrant, integrate the area of circular disks formed by the rotation, using the square of the function that represents x as a function of y, multiplied by pi, over the relevant bounds.
Step-by-step explanation:
To find the volume of a solid formed by rotating a region enclosed by curves in the first quadrant about the y-axis, you can use the method of disks or shells. However, since the student's question does not provide specific functions defining the region, we'll discuss the general approach using an example.
Let's consider a region bounded by the x-axis, y-axis, and the curve y=f(x) in the first quadrant. When this area is rotated about the y-axis, each slice perpendicular to the y-axis forms a disk with a radius equal to the x-value of the function f(x) at that y-value, and a thickness dy. The volume of each disk is πx2 dy, where x is a function of y, x=g(y). To find the total volume, integrate this expression with respect to y over the interval from y=0 to y=b, where y=b is where the curve intersects the y-axis.
Therefore, the volume V is given by the integral V = ∫0b π[g(y)]2 dy.
Without the specific equations for the region's boundary curves, we can't calculate an exact volume, but this approach will work for any such problem. For an actual computation, we would need the function g(y) that represents x as a function of y after solving the original equation for x in terms of y.