In this section, we will focus on using definite integration to determine the volumes of solids of revolution. These solids are created by revolving a region bounded by curves around an axis (x-axis or y-axis). We'll particularly explore methods to calculate the volume of solids formed by revolving regions between curves, like the volume of a solid formed when revolving the region between and around the x-axis.
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Volume of Revolution Between Two Curves
With Respect to x
Concept:
Finding the volume of revolution between two curves involves calculating the volumes formed by each curve around the x-axis and subtracting one from the other.
Procedure:
a. Ensure is the subject in the equations of the curves.
b. Apply the formula:
With Respect to y
Procedure:
a. Make the subject in the equations of the curves.
b. Use the formula:
Example Questions
Problem 1 :
Given: Curve and lines , .
Task: Calculate the volume when the shaded region is rotated 360° about the x-axis.
Solution:
1. Formula: Volume of revolution formula:
2. Integration:
simplifies to
3. Final Calculation:
Problem 2:
Given: The region between the curves and in the first quadrant.
Task: Find the volume of the solid formed when this region is revolved about the y-axis.
Solution:
1. Preparation: Express as the subject. Here, and .
2. Formula: Apply the formula for revolution about the y-axis:
3. Integration:
simplifies to
4. Final Calculation: