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find the volume of the solid generated by revolving the region bounded by the given curve and lines about the x-axis. y = 2sqrt(x)

User Jamila
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Final answer:

To calculate the volume of the solid formed by revolving y = 2sqrt(x) about the x-axis, we use the disk method, finding the volume by integrating the area of the circular cross-section, which depends on x. We integrate from the lower bound to the upper bound, multiplied by π, to find the total volume.

Step-by-step explanation:

To find the volume of the solid generated by revolving the curve y = 2sqrt(x) about the x-axis, we use the method of disks or cylindrical shells. For the disk method, we integrate the area of circular cross-sections perpendicular to the x-axis.

The area of a disk is given by πr², where r is the radius of the disk. In this case, the radius changes with x, and is equal to the function y = 2sqrt(x). Therefore, the area of a cross-section is π(2sqrt(x))² = 4πx.

The volume is found by integrating this area from the beginning to the end of the region of interest along the x-axis. Let's assume that the region is bounded by x=a and x=b. The volume V is then:

V = ∫_a^b 4πx dx = 4π ∫_a^b x dx

After integrating, we find V = 4π [x²/2]^b_a = 2π(b² - a²) and thus the volume of the created solid.

User Astex
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