2C ­ Upper and Lower Riemann Sums EXPORT.notebook
March 03, 2016
Evaluating Improper Integrals
Evaluating Improper Integrals
Divergent
Convergent
• Area under the curve cannot be approximated with a finite value. Why?
– Function was increasing
–
Function does not get smaller fast enough
• How do you know?
– Integrate function and answer = ∞
• Area under the curve can be approximated with a finite value. Why?
– Function does get smaller fast enough
• How do you know?
– Integrate function and get a non­∞ number
–
Power of x –
Power of x –
Comparison Test
–
Comparison Test
0 ≤ f(x) ≤ g(x)
0 ≤ f(x) ≤ g(x)
If an integral is convergent, then:
Review of Infinite Series
• What is the actual area under the curve?
64,16,4,1,...
– If you integrate function, your answer is the area
1.) Find the 20th term in the sequence.
– When using the power of x, you should be able to integrate if necessary to find the area even when you did not need it to prove convergence. 2.) Find the sum of the first 12 terms of the sequence.
– Comparison Test ­ How?
3.) Find the sum to infinity of the sequence.
0 ≤ f(x) ≤ g(x)
Upper and Lower Riemann Sums
Upper and Lower Riemann Sums
Decreasing Functions
Decreasing Functions
Upper Sums
Upper Sums
Lower Sums
Lower Sums
2C ­ Upper and Lower Riemann Sums EXPORT.notebook
Decreasing Functions
March 03, 2016
Decreasing Functions
Upper Sums
Upper Sums
Lower Sums
Lower Sums
Lower Sums
1
f(a+1)
1
f(a+2)
f(a+3)
Upper Sums
1
1
1
f(a+4)
1
f(a+1)
Decreasing Functions
Increasing Functions
1
f(a)
f(a+2)
1
f(a+3)
Ex1: Find the lower and upper sums (U and L) such that:
Upper Sums
Upper Sums
Hint: First identify whether the function is increasing or decreasing.
Lower Sums
Lower Sums
Ex2: a.) Find the lower and upper sums for and write the inequality using summation notation. You do not need to compute the actual values.
b.) Find the actual value of .