Download presentation
Presentation is loading. Please wait.
1
4.3 Riemann Sums and Definite Integrals
2
Objectives Understand the definition of a Riemann sum. Evaluate a definite integral using limits. Evaluate a definite integral using properties of definite integrals.
3
Riemann Sums When we find the area under a curve by adding rectangles, the answer is called a Riemann sum. subinterval partition The width of a rectangle is called a subinterval. The entire interval is called the partition. Subintervals do not all have to be the same size.
4
Riemann Sums Riemann Sums: Add areas of rectangles to estimate area. Rectangle widths don’t have to be the same. 3 basic types: –Left (use f(left endpoint) as height) –Right (use f(right endpoint) as height) –Midpoint (use f(midpoint) as height)
5
Norm of the Partitional subinterval partition If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by. As gets smaller, the approximation for the area gets better.
6
Definition of a Definite Integral If f is defined on [a,b] and the limit exists then f is integrable on [a,b] and the limit is denoted by Longest rectangle width 0 # rectangles ∞ discretecontinuous
![Definition of a Definite Integral If f is defined on [a,b] and the limit exists then f is integrable on [a,b] and the limit is denoted by Longest rectangle width 0 # rectangles ∞ discretecontinuous](http://images.slideplayer.com./32/9805927/slides/slide_6.jpg "Definition of a Definite Integral If f is defined on [a,b] and the limit exists then f is integrable on [a,b] and the limit is denoted by Longest rectangle width 0 # rectangles ∞ discretecontinuous")
7
Definite Integral Notation Leibniz introduced the simpler notation for the definite integral: Note that the very small change in x becomes dx.
8
Theorem 4.4: Continuity Implies Integrability If a function f is continuous of [a,b], then f is integrable on [a,b].
![Theorem 4.4: Continuity Implies Integrability If a function f is continuous of [a,b], then f is integrable on [a,b].](http://images.slideplayer.com./32/9805927/slides/slide_8.jpg "Theorem 4.4: Continuity Implies Integrability If a function f is continuous of [a,b], then f is integrable on [a,b].")
9
Example Evaluate the definite integral Remember: Why is it negative?
10
Theorem If f is continuous and nonnegative on [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
![Theorem If f is continuous and nonnegative on [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](http://images.slideplayer.com./32/9805927/slides/slide_10.jpg "Theorem If f is continuous and nonnegative on [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")
11
Example Consider the region bounded by the graph of f(x)=4x-x 2 and the x-axis.
12
Example Sketch the region corresponding to each definite integral, then evaluate each integral using a geometric formula.
13
Example Sketch the region corresponding to each definite integral, then evaluate each integral using a geometric formula.
14
Example Sketch the region corresponding to each definite integral, then evaluate each integral using a geometric formula.
15
Properties of Definite Integrals
16
More Properties
17
Homework 4.3 (page 272) #13-47 odd
Similar presentations
![Definition: the definite integral of f from a to b is provided that this limit exists. If it does exist, we say that is f integrable on [a,b] Sec 5.2:](/21/6236247/big_thumb.jpg "Definition: the definite integral of f from a to b is provided that this limit exists. If it does exist, we say that is f integrable on [a,b] Sec 5.2:>")
![Aim: Riemann Sums & Definite Integrals Course: Calculus Do Now: Aim: What are Riemann Sums? Approximate the area under the curve y = 4 – x 2 for [-1, 1]](/24/7261222/big_thumb.jpg "Aim: Riemann Sums & Definite Integrals Course: Calculus Do Now: Aim: What are Riemann Sums? Approximate the area under the curve y = 4 – x 2 for [-1, 1]>")
