Finding Volume of Solid with Perpendicular Rectangle Cross Sections

In summary, to find the volume of the solid enclosed by y=x^2 and y=3 with cross sections perpendicular to the y-axis as rectangles of height y^3, the base is 2y^1/2 and the height is y^3. Integrating the area of the cross section from 0 to 3 gives a volume of 4/9 * 3^(9/2).
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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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1. What is solid volume with integration?

Solid volume with integration is a mathematical concept used in calculus to find the volume of a solid shape. It involves using integration to sum up the volume of infinitesimally-thin slices of the shape to find the total volume.

2. How is solid volume with integration calculated?

To calculate the solid volume with integration, you need to first find the base area of the shape, then integrate the function that represents the shape's height or depth over the limits of the shape. This will give you the total volume of the solid shape.

3. What types of shapes can be measured using solid volume with integration?

Solid volume with integration can be used to find the volume of various shapes, such as cubes, spheres, cylinders, cones, and more complex shapes like pyramids and toroids.

4. What is the difference between solid volume with integration and traditional volume formulas?

The traditional volume formulas, such as those used for rectangular prisms or cylinders, are based on geometric shapes and do not take into account irregular or complex shapes. Solid volume with integration, on the other hand, can be used for any shape, making it a more versatile and accurate method for finding volume.

5. What real-world applications use solid volume with integration?

Solid volume with integration has many real-world applications, such as in engineering and physics to calculate the volume of objects and structures, in chemistry to determine the volume of substances, and in biology to measure the volume of organs or cells. It is also used in 3D modeling and computer graphics to create realistic representations of objects and environments.

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