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Find the volume of the solid generated by revolving the region in the 1st quadrant bounded by the coordinate axis, the curve y = e^(-x), and the line x = 1, about the:

y-axis
x-axis
The line x = 2
The line y = 2

1 Answer

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Final answer:

The volume of the solid created by revolving the given region around the y-axis is found using the disk method, integrating from x = 0 to x = 1. Other axes of revolution require adjustments to the respective integral formulas. Importantly, the volume of a sphere is 4/3πr3, different from the surface area formula.

Step-by-step explanation:

To find the volume of the solid generated by revolving the region bounded by the curve y = e(-x), the x-axis, and the line x = 1 about the y-axis, we use the disk method. The volume of the solid is calculated by integrating the area of the disks from x = 0 to x = 1. The formula for the volume of a disk is V = πr2h, where r is the radius and h is the height of the cylinder the disk belongs to.

For the revolution around other lines such as the x-axis, the line x = 2, and the line y = 2, we need to adjust our integral accordingly, e.g., by modifying the radius of the disks or shells and the range of integration.

Remember that the volume of a sphere is given by the formula 4/3πr3 and not 4πr2, which is the formula for the surface area of a sphere. To remember formulas correctly, it can be helpful to go back to basic shapes and build an understanding from there, such as knowing that a cylinder's volume is the area of the base times the height.

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