Find the volume of the solid generated by revolving the region bounded by y=x3,y=0 and x=2 about the y-axis using the washer method.
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Step-by-Step Solution
Step 1: Express x in terms of y
To use the washer method when revolving around the y -axis, we need to express the functions in terms of y. The given curve is y=x3. Taking the cube root of both sides, we get x=y1/3. The other boundary is x=2, which remains x=2.
Step 2: Determine the limits of integration
The region is bounded by y=0 (the x -axis) and x=2. When x=2, y=23=8. So, the limits of integration for y are from 0 to 8.
Step 3: Identify the outer and inner radii
When revolving about the y -axis, the outer radius R(y) is the distance from the y -axis to the outer boundary, which is x=2. The inner radius r(y) is the distance from the y -axis to the inner boundary, which is x=y1/3.
Step 4: Set up the integral for the volume
The formula for the volume using the washer method about the y -axis is V=π∫ab([R(y)]2−[r(y)]2)dy. Substituting the radii and limits, we get the integral.
Step 5: Evaluate the integral
Now, we evaluate the definite integral. We find the antiderivative of 4−y2/3, which is 4y−5/3y5/3. Then, we evaluate this expression at the upper and lower limits and subtract.