Created on 2024-06-20Asked by Noah Ramirez (Solvelet student)
Find the volume of the solid generated by revolving the region bounded by the curve y=x2 and the x-axis from x=0 to x=2 about the x-axis.
Solution
To find the volume of the solid generated by revolving the region bounded by the curve y=x2 and the x-axis from x=0 to x=2 about the x-axis, we use the method of disks or washers. 1. **Setup:** We will integrate along the x-axis from x=0 to x=2. The radius of each disk or washer at a given x is given by y=x2, and the height (or thickness) is dx. 2. **Volume of a disk or washer:** For a disk or washer at position x, the volume dV is given by: dV=πy2dx. 3. **Volume of the solid:** To find the total volume V, we integrate dV over the interval [0,2]: V=∫02π(x2)2dx. 4. **Evaluation of the integral:** V=π∫02x4dx=π[5x5]02=532π. 5. **Result:** The volume of the solid generated by revolving the region bounded by the curve y=x2 and the x-axis from x=0 to x=2 about the x-axis is 532π cubic units. Solved on Solvelet with Basic AI Model
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DefinitionA volume of revolution is a 3-dimensional object created when a two-dimensional circle is rotated about an axis perpendicular to the plane on which the object is created.For eg: The volume of a solid produced or generated by revolving or rotating the curve y=x2 y=x2 around the x-axis from x=0 to x=11 is found by the integral π∫01(x2)2dx=5π.
