chapter 6 the definite integral. § 6.1 antidifferentiation
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Chapter 6
The Definite Integral
§ 6.1
Antidifferentiation
Antidifferentiation
Definition Example
Antidifferentiation: The process of determining f (x) given f ΄(x)
If , then xxf 2
.2xxf
Finding Antiderivatives
EXAMPLEEXAMPLE
Find all antiderivatives of the given function.
89xxf
Theorems of Antidifferentiation
The Indefinite Integral
Rules of Integration
Finding Antiderivatives
EXAMPLEEXAMPLE
Determine the following.
dx
xxx
3
12 2
Finding Antiderivatives
EXAMPLEEXAMPLE
Find the function f (x) for which and f (1) = 3. xxxf 2
Antiderivatives in Application
EXAMPLEEXAMPLE
A rock is dropped from the top of a 400-foot cliff. Its velocity at time t seconds is v(t) = -32t feet per second.
(a) Find s(t), the height of the rock above the ground at time t.(b) How long will the rock take to reach the ground?
(c) What will be its velocity when it hits the ground?
§ 6.2
Areas and Riemann Sums
Area Under a Graph
Definition Example
Area Under the Graph of f (x) from a to b: An example of this is shown to the right
Area Under a Graph
In this section we will learn to estimate the area under the graph of f (x) from x = a to x = b by dividing up the interval into partitions (or subintervals),
each one having width where n = the number of partitions that
will be constructed. In the example below, n = 4.n
abx
A Riemann Sum is the sum of the areas of the rectangles generated above.
Riemann Sums to Approximate Areas
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Use a Riemann sum to approximate the area under the graph f (x) on the given interval using midpoints of the subintervals
4,22;2 nxxxf
The partition of -2 ≤ x ≤ 2 with n = 4 is shown below. The length of each subinterval is
.1
4
22
x
-2 2x1 x2 x3 x4
x
x
Riemann Sums to Approximate Areas
Observe the first midpoint is units from the left endpoint, and the midpoints themselves are units apart. The first midpoint is x1 = -2 + = -2 + .5 = -1.5. Subsequent midpoints are found by successively adding
CONTINUECONTINUEDD
x.1x
2/x2/x
midpoints: -1.5, -0.5, 0.5, 1.5
The corresponding estimate for the area under the graph of f (x) is
xfxfxfxf 5.15.05.05.1
xffff 5.15.05.05.1
5125.225.025.025.2
So, we estimate the area to be 5 (square units).
Approximating Area With Midpoints of Intervals
CONTINUECONTINUEDD
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-3 -2 -1 0 1 2 3
Riemann Sums to Approximate Areas
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Use a Riemann sum to approximate the area under the graph f (x) on the given interval using left endpoints of the subintervals
5,31;3 nxxxf
The partition of 1 ≤ x ≤ 3 with n = 5 is shown below. The length of each subinterval is
.4.05
13
x
3
x
x1 x2 x3 x4 x5
1 1.4 1.8 2.2 2.6
Riemann Sums to Approximate Areas
The corresponding Riemann sum is
CONTINUECONTINUEDD
xfxfxfxfxf 6.22.28.14.11
xfffff 6.22.28.14.11
12.154.06.22.28.14.11 33333
So, we estimate the area to be 15.12 (square units).
Approximating Area Using Left Endpoints
CONTINUECONTINUEDD
0
5
10
15
20
25
30
1 1.4 1.8 2.2 2.6 3 3.4
§ 6.3
Definite Integrals and the Fundamental Theorem
The Definite Integral
Δx = (b – a)/n, x1, x2, …., xn are selected points from a partition [a, b].
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.2 0.4 0.6 0.8 1 1.2
Calculating Definite Integrals
EXAMPLEEXAMPLE
SOLUTIONSOLUTION
Calculate the following integral.
1
05.0 dxx
The figure shows the graph of the function f (x) = x + 0.5. Since f (x) is nonnegative for 0 ≤ x ≤ 1, the definite integral of f (x) equals the area of the shaded region in the figure below.
10.5
1
Calculating Definite Integrals
The region consists of a rectangle and a triangle. By geometry,
5.05.01heightwidthrectangle of area
CONTINUECONTINUEDD
5.0112
1heightwidth
2
1 triangleof area
Thus the area under the graph is 0.5 + 0.5 = 1, and hence
.15.01
0 dxx
The Definite Integral
Calculating Definite Integrals
EXAMPLEEXAMPLE
Calculate the following integral.
1
1xdx
The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus
EXAMPLEEXAMPLE
Use the Fundamental Theorem of Calculus to calculate the following integral.
1
0
5.031 13 dxex x
Use TI 83 to compute the definite integral: 1) put f(x) into y1 and graph.2) 2nd trace 73) Enter lower limit and upper limit at the prompts.
Area Under a Curve as an Antiderivative
§ 6.4
Areas in the xy-Plane
Properties of Definite Integrals
Area Between Two Curves
Finding the Area Between Two Curves
EXAMPLEEXAMPLE
Find the area of the region between y = x2 – 3x and the x-axis (y = 0) from x = 0 to x = 4.
Finding the Area Between Two Curves
EXAMPLEEXAMPLE
Write down a definite integral or sum of definite integrals that gives the area of the shaded portion of the figure.
§ 6.5
Applications of the Definite Integral
Average Value of a Function Over an Interval
Average Value of a Function Over an Interval
EXAMPLEEXAMPLE
Determine the average value of f (x) = 1 – x over the interval -1 ≤ x ≤ 1.
Average Value of a Function Over an Interval
EXAMPLEEXAMPLE
(Average Temperature) During a certain 12-hour period the temperature at time
t (measured in hours from the start of the period) was degrees. What was the average temperature during that period?
2
3
1447 tt
Consumers’ Surplus
Consumers’ Surplus
EXAMPLEEXAMPLE
Find the consumers’ surplus for the following demand curve at the given sales level x.
20;10
3 xx
p
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