Applications of Integration

Area Between Curves

Integration can be used to find the area between two curves by integrating the difference of their functions over a specified interval.

• Key Concept: The area between two curves and from to is given by:

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• Example: Find the area bounded by and .

• Set up the integral by finding the intersection points and integrating the upper function minus the lower function.

Arc Length of a Curve

The arc length of a curve from to can be found using the following formula:

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• Key Steps: Compute the derivative , square it, add 1, take the square root, and integrate over the interval.

• Example: Find the arc length of on the interval .

Volume of Solids of Revolution

Volumes of solids generated by revolving a region around an axis can be found using the disk/washer or shell methods.

• Disk/Washer Method: For a region bounded by and revolved about the x-axis:

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• Shell Method: For a region revolved about the y-axis:

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• Example: Find the volume generated by revolving the region bounded by and about the x-axis.

Work Done by a Variable Force

When a force stretches or compresses a spring, the work done can be calculated using Hooke's Law and integration.

• Hooke's Law: , where is the spring constant.

• Work Formula:

$

• Example: A force of 50 pounds stretches a spring 6 inches. Find the work done in stretching the spring 1 foot (12 inches) from its natural position.

• First, find using , then set up the integral for work.

Practice Problems (from Exam)

1. Sketch the graph of the region bounded by the graphs of the functions and find its area.

2. Find the arc length of on the interval . Set up the integral, including simplification, but do not solve.

3. Find the volume of the solid generated by revolving the region bounded by the graphs of the functions about the x-axis using disks or washers. Set up the integral, but do not solve.

4. Find the volume of the solid generated by revolving the region bounded by the graphs of the functions about the x-axis using shells. Set up the integral, but do not solve.

5. A force of 50 pounds stretches a spring 6 inches in an exercise machine. Find the work done in stretching the spring 1 foot (12 inches) from its natural position.

Additional info: These problems cover core applications of integration, including area, arc length, volume, and work, which are essential topics in a Calculus II course.