Washers and disk graph5/2/2023 ![]() Thus, the cross-sectional area is given by The horizontal cross-sectional shape of our solid will be a disk, and the radius of each disk will be the value of the function at that point, ? = ? 9. Since we rotated this region about the ?-axis, the volume is given by Of revolution, but this time rotated about the ?-axis. Now, let’s look at an example where we determine the volume of a solid Thus, we have ? = 0 and ? = 2, and the volume of the solid of revolution is given by The solutions are given by ? = 0 and ? = 2, which can also be visualized from the plot of the region bounded by the curves. The limits of the integral, ? and ?, are determined by the vertical bounds of ? in the region, which in this case occurs when the curve ? = − ? 2 ? intersects the ?-axis or when ? = 0: The cross-sectional shape of our solid will be a disk, and the radius of each disk will be the value of the function at that point, ? = − ? 2 ? . Solid generated by the revolution of a region bounded by a parabola around the Now, let’s consider an example where we have to find the volume of Thus, the volume of the generated solid is Thus, we have ? = − 1 and ? = 4, and the volume of the solid of revolution is given by ![]() The upper bound is ? = 4 and the lower bound is determined by the intersection of the curves ? = √ ? 1 and ? = 0, which is ? = − 1 this can also be visualized from the plot of the region bounded by the curves. The limits of the integral, ? and ?, are determined by the vertical bounds of ? in the region. The vertical cross-sectional shape of our solid will be a disk, and the radius of each disk will be the value of the function at that point, ? = √ ? 1. Since we are rotating this region about the ?-axis, the volume is given by This will be determined using integration. Of a region, bounded by a radical function and other lines, about the In the next example, we will determine the volume generated by the revolution Thus, the volume of the solid of revolution is Thus, we have ? = 0 and ? = 3, and the volume of the solid of revolution is given by The limits of the integral, ? and ?, are determined by the vertical bounds of ? in the region bounded by the curves, 0 ≤ ? ≤ 3 this can also be visualized from the plot of the region bounded by the curves. The vertical cross-sectional shape of our solid will be a disk, and the radius, ?, of each disk will be the value of the function at that point, ? = ? 4. You can also get some more practice with the washer method here.Since we rotated the region about the ?-axis, the volume is given by Exampleįind the volume obtained by rotating the area bounded by \(y=x\) and \(y= \sqrt\)! Hopefully this has helped you with the washer method, but if there’s still a topic you’d like to learn about take a look at some of my other lessons and problem solutions about integrals. ![]() So let’s jump into an example and I’ll explain the difference as we go. Exactly as you would expect from the name, a washer is just a disk with a hole taken out of its center. ![]() You can think of the main difference between these two methods being that the washer method deals with a solid with a piece of it taken out. The washer method for finding the volume of a solid is very similar to the disk method with one small added complexity. Once you have the disk method down, the next step would be to find the volume of a solid using the washer method.
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