39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.


y = -2

Verified step by step guidance

1. First, identify the region R bounded by the curves: \(y = x^{2}\), \(x = 1\), and \(y = 0\). This region lies between \(x=0\) and \(x=1\) since \(y=0\) is the x-axis and \(y=x^{2}\) is a parabola opening upwards.
2. Since we are revolving the region about the line \(y = -2\), which is horizontal and below the x-axis, we will use the shell method with vertical shells. The shells will be formed by vertical slices of the region at position \(x\) with thickness \(dx\).
3. The height of each shell is the vertical distance between the curves \(y = x^{2}\) (top) and \(y = 0\) (bottom), so the height is \(h(x) = x^{2} - 0 = x^{2}\).
4. The radius of each shell is the distance from the slice at \(y\) to the axis of rotation \(y = -2\). Since the shell is at height \(y\) between \(0\) and \(x^{2}\), the radius is the vertical distance from \(y\) to \(-2\). However, because we are integrating with respect to \(x\), the radius is the distance from the shell at \(y = x^{2}\) to \(y = -2\), which is \(r(x) = x^{2} - (-2) = x^{2} + 2\).
5. The volume of the solid is given by the integral using the shell method formula: \(V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx\). Substitute the expressions for radius and height and the limits \(a=0\) and \(b=1\): \(V = 2\pi \int_{0}^{1} (x^{2} + 2)(x^{2}) \, dx\).

Key Concepts

Shell Method for Volume

The shell method calculates the volume of a solid of revolution by integrating cylindrical shells. Each shell's volume is found by multiplying its circumference, height, and thickness. This method is especially useful when the axis of rotation is parallel to the axis of the function being integrated.

Setting up the Shell Radius and Height

When using the shell method, the radius is the distance from the axis of rotation to the shell, and the height is the length of the shell segment. For rotation about y = -2, the radius is the vertical distance from y = -2 to a point y on the region, and the height corresponds to the horizontal length of the region at that y.

Region Boundaries and Limits of Integration

The region R is bounded by y = x², x = 1, and y = 0, defining the shape to be revolved. Understanding these boundaries helps determine the limits of integration and the expressions for height and radius in terms of the variable of integration, ensuring accurate volume calculation.