53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.
y = x²,y=2−x, and x=0, in the first quadrant; about the y-axis

Verified step by step guidance

1. First, identify the region R bounded by the curves: \(y = x^{2}\), \(y = 2 - x\), and \(x = 0\) in the first quadrant. Sketching these curves helps visualize the area and the solid formed when revolving around the y-axis.
2. Determine the points of intersection between \(y = x^{2}\) and \(y = 2 - x\) by setting \(x^{2} = 2 - x\). Solve this equation to find the x-values where the curves meet, which will serve as limits of integration.
3. Since the solid is generated by revolving the region around the y-axis, decide on the method to use: the shell method is often convenient when revolving around the y-axis and the functions are given in terms of x.
4. Set up the volume integral using the shell method formula: \(V = \int_{a}^{b} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx\). Here, the radius is the distance from the y-axis, which is \(x\), and the height is the vertical distance between the curves, \( (2 - x) - x^{2} \).
5. Write the integral explicitly as \(V = \int_{x=a}^{x=b} 2\pi x \left( (2 - x) - x^{2} \right) dx\), where \(a\) and \(b\) are the intersection points found earlier. This integral represents the volume of the solid formed by revolving the region around the y-axis.

Key Concepts

Setting up the region bounded by curves

Understanding the region R requires identifying the area enclosed by the curves y = x², y = 2 - x, and the line x = 0 in the first quadrant. This involves finding the points of intersection and sketching the region to visualize the limits of integration.

Methods of finding volumes of solids of revolution

Volumes generated by revolving a region around an axis can be found using methods like the disk/washer method or the shell method. Choosing the appropriate method depends on the axis of rotation and the shape of the region to simplify integration.

Volume calculation about the y-axis using the shell method

When revolving around the y-axis, the shell method is often convenient. It involves integrating cylindrical shells with radius equal to the x-value and height given by the difference of the functions, integrating with respect to x over the region's domain.