4.6. Practical Applications

Try these first

1. 1.

   What is the main purpose of numerical integration?

   Hint

   Think about scenarios where exact answers are hard to find.

2. 2.

   Name a numerical method used in numerical integration.

   Hint

   These methods are named after mathematicians.

3. 3.

   What is the main application of numerical integration?

   • To solve differential equations
   • To approximate definite integrals
   • To find exact solutions

   Hint

   Consider what numerical methods are designed for.

4. 4.

   True or False: Simpson's Rule can be used only if the number of intervals is odd.

   • True
   • False

   Hint

   Think about the requirements for applying Simpson's Rule.

5. 5.

   Suppose you have a force function acting on a vehicle, F(x) = 3x^2 - 5x + 2. Calculate the work done by this force over the interval [1, 3] using the Trapezoidal Rule with n=4.

   Hint

   Calculate stepwise according to the trapezoidal formula and integrate accurately.

6. 6.

   Imagine modeling the population growth of a bacteria colony with the differential equation dP/dt = 0.03P(1 - P/400) where P is the population. Use numerical integration to estimate the population after 10 days starting with P(0) = 50 using Simpson's Rule.

   Hint

   Focus on setting up the time intervals and using the growth rate as the function for integration.