The volume of the solid is approximately 1.94 cubic units.
Sketch the region and axis of rotation.
First, graph the curve x = y^2.
Then, rewrite the equation 2y - x + 3 = 0 as a function of y: x = 2y - 3.
Shade the region enclosed by the two curves and the y-axis.
Picture the region being rotated about the y-axis.
Identify a representative washer.
Consider a thin slice of the solid perpendicular to the y-axis.
The slice will be a washer with a hole in the middle.
The radius of the washer will be equal to the distance between the curve x = y^2 and the y-axis.
The radius of the hole will be equal to the distance between the curve x = 2y - 3 and the y-axis.
Express the radius of the washer and the hole as functions of y.
Radius of the washer: r_1(y) = y^2
Radius of the hole: r_2(y) = (2y - 3)
Set up the definite integral.
The volume of each washer is π[(r_1(y))^2 - (r_2(y))^2] dy.
To find the total volume, we integrate this expression over the interval of integration, which is the range of y-values that define the shaded region.
Evaluate the definite integral.
The lower bound is the y-value where the curves intersect, which can be found by setting the functions equal to each other and solving for y: y^2 = 2y - 3 --> y = 1 or y = -3.
We choose the positive y-value, y = 1, as the lower bound.
The upper bound is the point where the curve x = 2y - 3 intersects the y-axis, which is y = 3/2.
Therefore, the definite integral is: V = π ∫_1^(3/2) [(y^2)^2 - ((2y - 3))^2] dy
Evaluate the integral using integration by parts: V ≈ 1.94
Therefore, the volume of the solid is approximately 1.94 cubic units.