Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis.
x = 1 + (y − 6)^2, x = 17
The curves intersect at (17,2) and (17,10)
v = ∫[2,10] 2πrh dy
where r=y and h=17-x = 17-(1+(y-6)^2)
v = ∫[2,10] 2πy(17-(1+(y-6)^2)) dy = -2π∫[2,10] (y^3-12y^2+20y) dy
To use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the curves x = 1 + (y - 6)^2 and x = 17 about the x-axis, we need to follow these steps:
1. Sketch the region: First, we need to sketch the region bounded by the given curves. The curve x = 1 + (y - 6)^2 is a parabola opening to the right with a vertex at (1, 6) and the line x = 17 is a vertical line passing through the point (17, 0). The region lies between the x-axis and the curves.
2. Determine the limits of integration: To find the limits of integration for the cylindrical shells method, we need to determine the points where the curves intersect. At these intersection points, the lower bound will be the y-value of the point of intersection, and the upper bound will be the y-value of the curve that is highest at that x-value. In this case, the curves do not intersect, so we need to find the limits of integration based on the endpoints.
The y-values of the curves are obtained by solving the equations for y. For x = 1 + (y - 6)^2, we rewrite it as y = 6 ± √(x - 1). Taking the positive square root, we have y = 6 + √(x - 1). This equation gives us the upper bound for y. Since the region is bounded by the x-axis, the lower bound for y is 0.
3. Set up the integral: The formula to find the volume using the cylindrical shells method is given by V = ∫(2π * radius * height) dy, where the integral is evaluated from the lower bound to upper bound of y.
For each value of y within the limits of integration, the radius is x - axis, which is y. The height is the difference between the x-values of the curves at that y-value. The height can be found by subtracting the equation of the lower curve from the equation of the upper curve.
So, the integral to find the volume becomes V = ∫(2π * y * (17 - (1 + (y - 6)^2))) dy, evaluated from y = 0 to y = infinity.
4. Evaluate the integral: Solve the integral to find the volume. The indefinite integral is quite complex, so it would be best to use a computer algebra system or online integral calculator to find the exact result.
By following these steps, you should be able to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis using the method of cylindrical shells.