Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:


a. Apply the disk method and integrate with respect to y.

Verified step by step guidance

1. First, identify the region R bounded by the parabola \(y = 4 - x^{2}\) and the coordinate axes in the first quadrant. Since we are in the first quadrant, \(x \geq 0\) and \(y \geq 0\).
2. Express \(x\) as a function of \(y\) to set up the integral with respect to \(y\). Starting from \(y = 4 - x^{2}\), solve for \(x\): \(x = \sqrt{4 - y}\).
3. Since the solid is formed by revolving the region around the y-axis, the cross-sectional disks will be horizontal slices perpendicular to the y-axis. The radius of each disk is the \(x\)-value at that \(y\), which is \(r(y) = \sqrt{4 - y}\).
4. The area of each disk is \(A(y) = \pi [r(y)]^{2} = \pi (4 - y)\). The volume is found by integrating these areas along the \(y\)-axis from the lowest to the highest \(y\)-value in the region, which are \(y=0\) to \(y=4\).
5. Set up the volume integral using the disk method: \(V = \int_{0}^{4} \pi (4 - y) \, d y\). This integral will give the volume of the solid when evaluated.

Key Concepts

Disk Method for Volume

The disk method calculates the volume of a solid of revolution by slicing the solid perpendicular to the axis of rotation into thin disks. Each disk's volume is approximated by π(radius)²(thickness), and integrating these volumes over the given interval yields the total volume.

Rewriting Functions in Terms of y

When integrating with respect to y, it is necessary to express the function x in terms of y. This involves solving the given equation y = 4 - x² for x, which allows setting up the integral with y as the variable of integration.

Bounds of Integration in the First Quadrant

The region is bounded by the parabola and coordinate axes in the first quadrant, so the limits of integration correspond to the y-values where the region exists, typically from y = 0 up to the maximum y-value on the parabola within the first quadrant.