Answer:
Explanation:
To use the method of cylindrical shells to find the volume of the region, we need to set up an integral using the circumference of the cylinder and its height. The circumference is given by 2πr, where r is the radial distance from the y-axis, and the height is given by the difference in y-coordinates of the two curves at a given x-coordinate.Let's first find the equation for r in terms of x:y = 2 + x - x^2
y + x = 2Solving for x:x = 2 - y
r = y + x
r = 2 + y - y
r = 2So, the equation for r is simply 2.Next, we'll find the limits of the integral:The x-coordinate of the first intersection point is 1, and the x-coordinate of the second intersection point is 0. So, the limits of the integral will be from x = 0 to x = 1.Finally, we can set up the integral:V = ∫_0^1 (2πr)(y2 - y1) dx
= ∫_0^1 (2π)(2)(y2 - y1) dx
= 2π ∫_0^1 (y2 - y1) dx
= 2π ∫_0^1 (2 + x - x^2 - (2 - x)) dx
= 2π ∫_0^1 (x^2 + 3x + 2) dxEvaluating this definite integral gives us the volume generated by rotating the region bounded by the curves about the y-axis:V = 2π [x^3/3 + 3x^2/2 + 2x]_0^1
= 2π [(1^3)/3 + 3(1^2)/2 + 2(1) - (0^3)/3 - 3(0^2)/2 - 2(0)]
= 2π [(1/3) + 3/2 + 2 - 0]
= 2π (13/6)
= 13π/3 cubic units.