Finding Volume of Solid with Perpendicular Rectangle Cross Sections

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SUMMARY

The volume of the solid with perpendicular rectangular cross sections can be calculated using the region enclosed by the curves y=x^2 and y=3. The base of the rectangular cross section is determined to be 2y^(1/2), and the height is y^3. The area of the cross section is thus 2y^(7/2). Integrating this area from 0 to 3 yields the volume, which results in 4/9 * 3^(9/2). This calculation has been verified by multiple participants in the discussion.

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  • Knowledge of polynomial functions and their properties
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Homework Statement



The base is the region enclosed by y=x^2 and y=3. The cross sections perpendicular to the y-axis are rectangles of height y^3.

Use the information to solve for the volume of the solid.

Homework Equations





The Attempt at a Solution



I tried to find the base and height of the cross sections in terms of y, since the parabola is 2y^1/2 across for any height that is the base of the rectangular cross section is 2y^1/2 and the height will be y^3. This means the area of the cross section is the product of (2y^1/2)*y^3 =2*y^(7/2) then this integrated from 0 to 3 should give me the volume? I ended up getting 48/9 * 3^1/2 or 4/9 * 3^(9/2) could anyone verify this?
 
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