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6.1 - Estimating with Finite Sums
6.3 - Definite Integrals and Antiderivatives
6.4 - The Fundamental Theorem of Calculus
- LRAM
- Sum of rectangle areas, heights are given by f(a) where a is the left endpoint of each subinterval
- Over-estimate if f(x) is decreasing
- Under-estimate if f(x) is increasing
- RRAM
- Sum of rectangle areas, heights are given by f(b) where b is the right endpoint of each subinterval
- Under-estimate if f(x) is decreasing
- Over-estimate if f(x) is increasing
- MRAM
- Sum of rectangle areas, height = f(c) is given by using x = c, which is the midpoint of [a, b] where a → b is the width of the rectangle
- Difficult to tell if over or under estimation, but closer approx than LRAM, RRAM
- Need to be able to do all given
- Graph
- Equation
- Table
- Sum of trapezoidal areas, base is = width of subintervals, heights are given by f(a) and f(b) if [a , b] is the subinterval length
- Each height is used twice, EXCEPT the first and last (endpoints)
- Closer than any RAM estimation
- Over-estimate if f(x) is concave up
- Under-estimate if f(x) is concave down
- Using geometry to calculate the exact area under a curve and above the x-axis (triangles, circles - (quarter, semi, etc), trapezoids, rectangles)
- Area that is under the x-axis and above a curve becomes negative
- How to use Graphing Calculator to find definite integrals (MATH → 9:fnINT( → plug in bounds and evaluate)
6.3 - Definite Integrals and Antiderivatives
- Properties for Definite Integrals
- Out of order, moving right to left: if lower bound is > upper bound, make integral negative
- The integral from a → a = 0
- Can take apart integrals and add / subtract pieces separately
- Can take out a constant multiple, evaluate the integral, then multiply it back in
- Can add two pieces to make a whole integral (integral from a → c = integral from a → b + integral from b → c)
- Reverse Power Rule
- ADDING A “+ C” FOR INDEFINITE INTEGRALS (no lower/upper bound)
- Solving for C, given initial condition and plugging in for x, y after finding indefinite integral
- using definite integrals to find distance and displacement
6.4 - The Fundamental Theorem of Calculus
- taking the derivative of an integral
For help with reimann sums, watch-midnight tutor reimann sums:
http://www.youtube.com/watch?v=YHYT3HlL1uA
for help with fundamental theorem of calculus
http://patrickjmt.com/fundamental-theorem-of-calculus-part-1/
http://www.youtube.com/watch?v=YHYT3HlL1uA
for help with fundamental theorem of calculus
http://patrickjmt.com/fundamental-theorem-of-calculus-part-1/