SUMMARY
The volume of water in a swimming pool can be calculated using a double integral based on the depth function f(x,y) = 2sin(x/20 - 7) - 3cos((x-3)/5) + 8 and the side equations y(x) = 10 - (x-10)²/10 and y(6) = (x-10)²/20 - 5. To set up the integral, graph the parabolas representing the sides of the pool and determine the length of a rectangle perpendicular to the x-axis, which is the difference between the two y-values. The volume can be computed by integrating the product of the rectangle's length and depth function over the defined limits.
PREREQUISITES
- Understanding of double integrals in calculus
- Familiarity with trigonometric functions and their properties
- Knowledge of parabolic equations and their graphical representations
- Ability to perform definite integrals
NEXT STEPS
- Learn how to graph parabolic functions and their intersections
- Study the application of double integrals in calculating volumes
- Explore the use of trigonometric identities in integrals
- Practice setting up and solving definite integrals with variable limits
USEFUL FOR
Students in calculus courses, educators teaching integral calculus, and anyone interested in applying mathematical concepts to real-world problems involving volume calculations.