Aim: What are Riemann Sums?. Do Now:. Approximate the area under the curve y = 4 – x 2 for [-1, 1] using 4 inscribed rectangles. Devising a Formula. Using left endpoint to approximate area under the curve is. the more rectangles the better the approximation. lower sum.
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Approximate the area under the curve y = 4 – x2 for [-1, 1] using 4 inscribed rectangles.
the more rectangles the better the approximation
yn - 1
the exact area?
take it to the limit!
yn - 1
yn - 2
left endpoint formula
yn - 1
right endpoint formula
Partition the interval into n subintervals not necessarily of equal length.
a = x0 < x1 < x2 < . . . < xn – 1< xn = b
Δxi = xi – xi – 1
- arbitrary/sample points for ith interval
Let f be defined on the closed interval [a, b], and let Δ be a partition of [a, b] given by a = x0 < x1 < x2 < . . . . < xn – 1 < xn = b,
where Δxi is the length of the ith subinterval. If ci is any point in the ith subinterval, then the sum
is called a Riemann sum for f for the partition Δ
largest subinterval – norm - ||Δ|| or |P|
equal subintervals – partition is regular
converse not true
Evaluate the Riemann Sum RP for
f(x) = (x + 1)(x – 2)(x – 4) = x3 – 5x2 + 2x + 8 on the interval [0, 5] using the Partition P with partition points 0 < 1.1 < 2 < 3.2 < 4 < 5 and corresponding sample points
If f is defined on the closed interval [a, b] and the limit
exists, the f is integrable on [a, b] and the limit is denoted by
The limit is called the definite integral of f from a to b. The number a is the lower limit of integration, and the number b is the upper limit of integration.
Definite integral is a number
Indefinite integral is a family of functions
If a function f is continuous on the closed interval [a, b], then f is integrable on [a, b].
not the area
The Definite Integral as Area of Region
If f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded by the graph of f, the x-axis and the vertical lines x = a and x = b is given by
Sketch & evaluate area region using geo. formulas.
A = lw
take the limit n