Find the volume of the solid obtained by rotating the region bounded
y = 16 x and y = 2 x^2 about y =0
Find the volume of the solid obtained by rotating the region bounded about the x-axis by y=4x^2, x =1, and y = 0
Find the volume of the solid generated by rotating the region bounded by
y = \sin^2(x) and y = 0
between 0 and pi about the x-axis
khalid
To find the volume of a solid obtained by rotating a region around an axis, you can use the method of cylindrical shells or the method of disks/washers, depending on the shape of the region.
For the first question:
To find the volume of the solid obtained by rotating the region bounded by y = 16x and y = 2x^2 about y = 0, we can use the method of cylindrical shells.
1. First, sketch the region bounded by the given curves to get an idea of what it looks like.
2. Determine the limits of integration. To find the volume, we integrate with respect to y since we are rotating about the y-axis. The region starts from y = 0 and ends at the intersection point of the two curves.
To find the intersection point, set the two equations equal to each other:
16x = 2x^2
Simplifying:
2x^2 - 16x = 0
2x(x - 8) = 0
This gives us two solutions: x = 0 and x = 8. We are only interested in the positive x-values, so the intersection point is x = 8.
3. Set up the integral using the formula for the volume of a cylindrical shell:
V = ∫ 2πxy dy, where x is the radius of the shell, and y represents the height of each shell.
Since x = y/16 for the curve y = 16x, and x = √(y/2) for the curve y = 2x^2, we have:
V = ∫[0 to 8] 2π(y/16)(√(y/2) - y/16) dy
4. Evaluate the integral using appropriate integration techniques, such as u-substitution or integration by parts.
Once you calculate this integral, you will find the volume of the solid obtained by rotating the region bounded by y = 16x and y = 2x^2 about y = 0.
For the second and third questions:
The volume of the solid obtained by rotating a region bounded about the x-axis can be found using the method of disks or washers.
1. Sketch the region bounded by the given curves to visualize the shape of the region.
2. Determine the limits of integration. To find the volume, we integrate with respect to x since we are rotating about the x-axis. The limits of integration are the x-values where the curves intersect or the boundaries specified in the question.
3. Set up the integral using the appropriate formula for volume:
V = ∫ πr^2 dx, where r is the radius of each disk or washer and dx represents the width of each disk or washer.
For the second question, the region is bounded by y = 4x^2, x = 1, and y = 0. In this case, the radius is the x-value, and the height or width is dy.
So, the integral to find the volume would be:
V = ∫[0 to 1] π(1)^2 dy
Evaluate this integral to find the volume of the solid obtained by rotating the region bounded by y = 4x^2 about the x-axis.
For the third question, the region is bounded by y = sin^2(x) and y = 0 between 0 and π. In this case, the radius is the y-value, and the height or width is dx.
So, the integral to find the volume would be:
V = ∫[0 to π] π(sin^2(x))^2 dx
Evaluate this integral to find the volume of the solid generated by rotating the region bounded by y = sin^2(x) and y = 0 between 0 and π about the x-axis.
Remember to carry out the necessary calculations and simplifications to obtain the final volume.