Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by y=2x^2 and y=x^3 about the x-axis.

The method of disks/rings can be used to determine the volume of a solid obtained by rotating a region bounded by two equations about the x-axis. In this particular case, the equations are y=2x^2 and y=x^3, with the region being rotated from x=0 to x=1. To calculate the volume using this method, we need to first find an equation for the area of each disk slice that is created when rotating around x-axis. This can be found by taking the integral between two points (in this case 0 and 1) of a function representing half circle with radius equal to difference between functions (f(x)-g(x)). The result will be pi times integral over range multiplied by square of difference in functions. So, in our example it would look like π∫_0^1 (2×2 – x[sup]3)[sup]2 ~dx = 4π/3.

Use the method of disks/rings to determine the volume of the solid obtained by rotating the region bounded by y=2x^2 and y=x^3 about the x-axis.

Now that we have found an equation for area of slices we can use formula for volume obtained by rotation which involves multiplying number rings, height each ring given length 2pi*r*h where r is radius at certain point h is distance from axis top bottom function Finally multiplies sum all these values together get total volume

In conclusion Method Disks Rings useful calculating volumes solid objects obtained rotation certain area around horizontal vertical axes In order calculate must first find expression area individual slices Then follows multiplying this number height Each slice along circumference Lastly multiply resulting value entire length obtain volume required Solving problem above yields 4π/3 as answer

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