Applications of Integration

Area Between Curves

Integration can be used to find the area between two curves by subtracting the lower function from the upper function over a given interval.

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

• Example: Find the area bounded by and from to .

• Application: Useful for determining the net area between two physical or theoretical boundaries.

Arc Length of a Curve

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

• Key Formula:

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

• Application: Used in engineering and physics to determine the length of wires, roads, or other curved objects.

Volume of Solids of Revolution

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

• Disk/Washer Method: For revolving around the x-axis:

• Shell Method: For revolving around the y-axis:

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

• Application: Used to determine the volume of objects with rotational symmetry, such as tanks or bottles.

Work Done by a Variable Force

Work done by a variable force, such as stretching a spring, can be calculated using integration.

• Hooke's Law: The force required to stretch a spring is proportional to the displacement: .

• Work Formula:

• Example: A force of 50 pounds stretches a spring 9 inches. Find the work done in stretching the spring 1 foot (12 inches) from its natural position.

• Application: Used in physics and engineering to calculate energy required for mechanical tasks.

Summary Table: Applications of Integration

Additional info: These problems are typical of a Calculus II course, focusing on the applications of definite integrals in geometry and physics.