Post on 25-Aug-2020
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Riemann Sums and Definite Integrals4.3
• Understand the definition of a Riemann sum.
• Evaluate a definite integral using limits & Riemann Sums.
• Evaluate a definite integral using geometric formulas
• Evaluate a definite integral using properties of definite integrals.
Objectives
Riemann SumsIn mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It may also be used to define the integration operation.
The method was named after German mathematician Bernhard Riemann. Some examples are Upper Sums, Lower Sums, and Midpoint Sums like we learned about in Section 4.2.
Example – A Partition with Subintervals of Unequal Widths
Consider the region bounded by the graph of and the xaxis for 0 ≤ x ≤ 1, as shown
Notice that the rectangles are notthe same width. You don’t have to have equal widths to do a Riemann Sum, (but it is easier to do if the subintervals have equal widths).
Riemann Sums
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Definite IntegralsBasically, as we divide a region into an infinite number of rectangles, each having a width of , we get infinitely close to the actual area of the region. This is called the definite integral and is denoted by:
where a and b are upper and lower limits.
Definite Integrals
Basically, if a function is continuous, then you can integrate it, (technically).
Definite Integrals as AreaAs an example of Theorem 4.5, consider the region bounded by the graph of f(x) = 4x – x2 and the xaxis, as shown in Figure 4.22.
Because f is continuous and nonnegative on the closed interval [0, 4], the area of the region is
Definite Integrals as Area
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Definite IntegralsBecause the definite integral in the example below is negative, it does not represent the area of the region shown.
Definite integrals can be positive, negative, or zero.
For a definite integral to be interpreted as an area, the function f must be continuous and nonnegative on [a, b].
You can evaluate a definite integral in more than one way:
You can use the limit definition & Reimann Sums You can check to see whether the definite integral represents the area of a common geometric region such as a rectangle, triangle, or semicircle.
Definite Integrals
Example – Areas of Common Geometric Figures
Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula.
a.
b.
c.
Properties of Definite IntegralsThe definition of the definite integral of f on the interval [a, b] specifies that a < b. It is, however, convenient to extend the definition to cover cases in which a = b or a > b. Geometrically, the following two definitions seem reasonable.
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Example – Evaluating Definite Integrals
a. Because the sine function is defined at x = π, and the upper and lower limits of integration are equal, you can write
b. The integral has a value of
so what is
The larger region can be divided at x = c into two subregions. It follows that the area of the larger region is equalto the sum of the areas of the two smaller regions.
cont’dExample – Evaluating Definite Integrals
Example – Using the Additive Interval Property Properties of Definite Integrals
Note that Property 2 of Theorem 4.7 can be extended to cover any finite number of functions. For example,
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Example – Evaluation of a Definite Integral
Evaluate using each of the following values.If f and g are continuous on the closed interval [a, b] and
0 ≤ f(x) ≤ g(x) for a ≤ x ≤ b, the following properties are true.
First, the area of the region bounded by the graph of f andthe xaxis (between a and b) must be nonnegative.
Properties of Definite Integrals
Second, this area must be less than or equal to thearea of the region bounded by the graph of g and the xaxis (between a and b ), as shown in Figure 4.25. These two properties are generalized in Theorem 4.8.
Properties of Definite Integrals The Fundamental Theorem of Calculus
4.4
• Evaluate a definite integral using the Fundamental Theorem of Calculus.• Find the average value of a function over a closed interval.• Understand and use the Second Fundamental Theorem of Calculus.
Objectives
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The Fundamental Theorem of Calculus
We can now evaluate a definite integral using Riemann Sums (or the trapezoidal rule),
& we can use geometric formulas, but what are the problems with using these two methods?
The Fundamental Theorem of Calculus
The two major branches of calculus are differential calculus and integral calculus. At this point, these two problems might seem unrelated—but there is a very close connection.
The connection was discovered independently by Isaac Newton and Gottfried Leibniz and is stated in a theorem that is appropriately called the Fundamental Theorem of Calculus.
The Fundamental Theorem of Calculus The Fundamental Theorem of CalculusGUIDELINES FOR USING THE FUNDAMENTAL THEOREM OF CALCULUS
* Provided you can find an antiderivative of f, you now have a way to evaluate a
definite integral without having to use the limit of a sum.
*
* Example:
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Why don't we need the "+C" any more? Example – Evaluating a Definite IntegralEvaluate each definite integral.
Example
Remember these problems from Section 4.3? We used geometry to find the areas let's do them now using the fundamental theorem of calculus:
a.
b.
Consider the region bounded by the graph of f(x) = 4x – x2 and the xaxis, as shown.Find the area of the shaded region.
Definite Integrals
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Average Value of a FunctionThe area of the region under the graph of f is equal to the area of the rectangle whose height is the average value.
Average value is like average height.
Average Value of a Function
(ba) is just the total width of the area we are integrating.
Example – Finding the Average Value of a Function
Find the average value of f(x) = 3x 2 – 2x on the interval [1, 4]. The definite integral of f on the interval [a, b] is defined using the constant b as the upper limit of integration and x as the variable of integration.
A slightly different situation may arise in which the variable x is used in the upper limit of integration.
To avoid the confusion of using x in two different ways, t is temporarily used as the variable of integration.
The Second Fundamental Theorem of Calculus
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The Second Fundamental Theorem of Calculus The Second Fundamental Theorem of Calculus
But what if we are doing the derivative of an integral. Then what would happen?
If we are just told to integrate, we evaluate using the First Fundamental Theorem of Calculus:
This result is generalized in the following theorem, called the Second Fundamental Theorem of Calculus.
The Second Fundamental Theorem of Calculus
Remember, this only works if you are taking the derivative of an integral, not the other way around, (integral of a derivative). Also, there must be a constant for the lower limit and x in the upper limit.
Example – Using the Second Fundamental Theorem of Calculus
Evaluate
Solution:Note that is continuous on the entire real line. So, using the Second Fundamental Theorem of Calculus, you can write:
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Examples:
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2.
3.
The Second Fundamental Theorem of CalculusRemember we said there must be a constant for the lower limit and an x in the upper limit to use the Second Fundamental Theorem of Calculus. It turns out that you can also use the theorem when the lower limit is a constant and the upper limit is a function of x. The only difference is that we plug in the function of x for t (instead of just the x), and we also multiply by the derivative of the function we plugged in. Here is an example:
Examples:
1.
2.
3.