## Example 2 | Volumes of Solids of Revolution

**Example 2**

Find the volume generated when the area in Example 1 will revolve about the *y*-axis.

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**Example 2**

Find the volume generated when the area in Example 1 will revolve about the *y*-axis.

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**Example 2**

Find the volume generated when the area in Example 1 will revolve about the *y*-axis.

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**Example 3**

Find the area bounded by the curve *x* = *y*^{2} + 2*y* and the line *x* = 3.

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**Example 1**

Find the area bounded by the curve *y* = 9 - *x*^{2} and the *x*-axis.

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The solid generated by rotating a plane area about an axis in its plane is called a **solid of revolution**. The volume of a solid of revolution may be found by the following procedures:

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There are two methods for finding the area bounded by curves in rectangular coordinates. These are...

- by using a horizontal element (called strip) of area, and
- by using a vertical strip of area.

The strip is in the form of a rectangle with area equal to length × width, with width equal to the **differential element**. To find the total area enclosed by specified curves, it is necessary to sum up a series of rectangles defined by the strip.

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Area, moment of inertia, and radius of gyration of parabolic section

**Situation**

Given the parabola 3x^{2} + 40y – 4800 = 0.

Part 1: What is the area bounded by the parabola and the X-axis?

A. 6 200 unit^{2}

B. 8 300 unit^{2}

C. 5 600 unit^{2}

D. 6 400 unit^{2}

Part 2: What is the moment of inertia, about the X-axis, of the area bounded by the parabola and the X-axis?

A. 15 045 000 unit^{4}

B. 18 362 000 unit^{4}

C. 11 100 000 unit^{4}

D. 21 065 000 unit^{4}

Part 3: What is the radius of gyration, about the X-axis, of the area bounded by the parabola and the X-axis?

A. 57.4 units

B. 63.5 units

C. 47.5 units

D. 75.6 units