Final answer:
To set up the volume of the solid obtained by rotating the region bounded by the curve 2y = x² and the line 5x - 2y - 6 = 0 about the line x = -1, we can use the method of cylindrical shells.
Step-by-step explanation:
To set up the volume of the solid obtained by rotating the region bounded by the curve 2y = x² and the line 5x - 2y - 6 = 0 about the line x = -1, we can use the method of cylindrical shells. Since the curve and the line intersect at two points, we need to determine the limits of integration. We can do this by solving the equations 2y = x² and 5x - 2y - 6 = 0 simultaneously. Substituting x² for y in the second equation, we get 5x - 2x² - 6 = 0. Solving this quadratic equation, we find the x-coordinates of the two points of intersection. Let's call them x₁ and x₂. The limits of integration will be x₁ and x₂.
The equation of the line x = -1 gives us the radius of the cylindrical shells. The distance from the line x = -1 to any point (x, y) on the curve 2y = x² is x + 1. The height of each cylindrical shell is the difference between the y-coordinates of the curve and the line at a given x-value. The volume of each cylindrical shell is given by the formula V = 2πrhΔx, where r is the radius, h is the height, and Δx is the width of the shell. To set up the integral, we need to express r, h, and Δx as functions of x. In this case, r = x + 1, h = (x² - 2y) - (-1) = x² + 2, and Δx = dx. Therefore, the volume integral is ∫(x₁ to x₂) 2π(x + 1)(x² + 2) dx.