Question

Find the volume of the solid generated by revolving the region bounded by the given line and curve about the x-axis. y= V9-x?,y=0 Find the volume of the solid generated by revolving the region bounded by the given curve and lines about the x-axis. y = 2VX, y=6, x=0 Find the volume of the solid generated by revolving the region bounded by the graphs of y-2x + 1 and y - 2x + 11 about the x-axis. Find the volume of the solid generated by revolving the following region about the y-axis. The region in the first quadrant bounded above by the parabola y=x, below by the x-axis, and on the right by the line x = 2.

Answer 1

To calculate the volume of a solid generated by revolving a region around an axis, integral calculus can be utilized with either the disc method or the shell method, depending on the region's shape.

Step-by-step explanation:

The volume of a solid generated by revolving a region around an axis can be computed using the disc method or shell method in integral calculus. The choice of method often depends on the symmetry and complexity of the region. For example, when revolving the region bounded by the line y = 2Vx, the line y = 6, and the line x = 0 around the x-axis, we would use the disc method and set up an integral with limits of integration defined by the x-values that correspond to where the curves intersect. Similarly, to find the volume of a solid generated by revolving the region between y = 2x + 1 and y = 2x + 11 around the x-axis, we would use the washer method due to the presence of two different functions. For the solid generated by revolving the region in the first quadrant bounded by the parabola y = x, the x-axis, and the line x = 2 around the y-axis, the shell method often proves more convenient. The key is to find the appropriate method to simplify the integral evaluation for calculating the volume.