Answer:
Explanation:
To find the volume created by revolving the area between the curve y = x^2 + 1 and the line y = 10 about the line y = 10, we can use the method of cylindrical shells.
First, we need to find the limits of integration. Since the curve and the line intersect at y = 10 and y = x^2 + 1 = 10, we have x^2 = 9, which gives us x = ±3. So, our limits of integration are from x = -3 to x = 3.
Next, we need to find the radius of each cylindrical shell. The distance between the line y = 10 and the curve y = x^2 + 1 is simply 10 - (x^2 + 1) = 9 - x^2.
Finally, we can set up the integral to find the volume:
V = ∫ from -3 to 3 2π(9 - x^2)(x) dx
We multiply by 2π since we are revolving around the line y = 10 and we integrate with respect to x.
Evaluating the integral, we get:
V = 2π ∫ from -3 to 3 (9x - x^3) dx
V = 2π [(4.5x^2 - 0.25x^4) from -3 to 3]
V = 2π [(4.5(3)^2 - 0.25(3)^4) - (4.5(-3)^2 - 0.25(-3)^4)]
V = 2π [27 - 6.75]
V = 42.39π
Therefore, the volume created by revolving the area between the curve y = x^2 + 1 and the line y = 10 about the line y = 10 is approximately 42.39π cubic units.