Focus on summation formulas essential for calculus, especially for calculating the area of a region.
Evaluation of sums using sigma notation.
Summation of a Constant
Formula: If (i) runs from 1 to (n) for a constant (c), then the sum equals (c \times n).
Example: Sum of 8 over 4 terms is (8 \times 4 = 32).
Example Problems:
(c = 7), (n = 8): (7 \times 8 = 56).
(c = 9), (n = 5): (9 \times 5 = 45).
Summation of Integers
Formula: Sum of integers (i) = (\frac{n(n+1)}{2}).
Example: Sum from 1 to 5 is (\frac{5(5+1)}{2} = 15).
Confirmation with actual addition: 1 + 2 + 3 + 4 + 5 = 15.
Sum of Multiples
Example: Sum from 1 to 4 of 6(i) terms.
Calculation: 6 + 12 + 18 + 24 = 60.
Using formula: (6 \times \frac{4(4+1)}{2} = 60).
Sum of Linear Expressions
Example: Sum from 1 to 5 of (7i - 3).
Sequence: 4, 11, 18, 25, 32.
Sum: 90.
Using formulas:
Break into parts: (7 \times \frac{5(5+1)}{2} - 3 \times 5).
Result: 105 - 15 = 90.
Practicing with Adjustments
Example: Sum from 1 to 8 of (9i + 7).
Calculated: 380.
Using formula: Confirmed via actual adding each term.
Summation of Squares
Formula: Sum of squares (i^2 = \frac{n(n+1)(2n+1)}{6}).
Example: Sum of squares to 6:
Calculation: 1, 4, 9, 16, 25, 36 = 91.
Formula: (\frac{6(6+1)(2 \times 6+1)}{6} = 91).
Summation of Cubes
Formula: Sum of cubes (i^3 = \frac{n^2(n+1)^2}{4}).
Example: Sum of cubes to 4:
1, 8, 27, 64 = 100.
Formula: ((4^2 \times 5^2)/4 = 100).
Practice Examples
Example: Sum of (i(i^2 + 4i)).
Distribute and separate: (i^3 + 4i^2).
Evaluate each part:
(\frac{n^2(n+1)^2}{4} + 4\frac{n(n+1)(2n+1)}{6}).
Result for (n=5): 445.
Final Practice
Example: Sum from 1 to 9 of ((i-1)^2).
Expand and separate:
(i^2 - 2i + 1).
Evaluate using formulas for each term.
Result: 204.
Conclusion
Understanding and applying different summation formulas can simplify evaluations in calculus, especially for identifying areas under curves and other advanced calculations.