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EXAMPLES FOR EXAM IV
| Antiderivative Examples
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| Area Approximation
Find
the left, right, and midpoint approximations to the area of the region
bounded by the graphs of the relations given using four approximating
rectangles. The figure at the left below shows the left and right
approximating rectangles. The figure at the right below shows the
midpoint approximating rectangles.
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| Approximate the area of the
region bounded by the graphs of the given equations with n = 100
using a TI and compute the exact area using the limit process.
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| Follow this
link to a nice Section 4.2 Example by fellow Seminole State math
instructor Scott Rickman. The example uses DPGraph to demonstrate
approximating the area under a curve by the left, right, and midpoint
approximations. The example is also notable for the very clever way
he has done it. |
Area Using
the Fundamental Theorem of Calculus (Powerpoint
Presentation of the 4 Examples Below)
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Compute the area of the region bounded by the
graphs of the given equations.
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| Compute the area of the region
bounded by the graphs of the given equations.
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| Compute the area of the region
bounded by the graphs of the given equations.
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| Compute the area of the region
bounded by the graphs of the given equations.
Note: This is the area of a
trapezoid  |
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| Area Example
Can
you tell what would happen as the right side boundary (x = 10)
moves
farther and farther to the right resulting in the upper limit of
integration getting larger and larger?
EC: Find the limiting area as B approaches infinity if the right
boundary
is
x = B. |
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| Area Approximation
You will need to approximate x-intercepts in
order to find the limits of integration. You will need to do a
numerical integration on your calculator to approximate the definite
integral that represents the area.
Maple Worksheet
TI-84 Audio/Video Presentation |
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Here is a
Maple Worksheet investigating the area of the
region between the x-axis and the graph of f(x) = x3 - 2x2
- 3x + 10
and between
the graphs of x = -1 and x = 3.
| Other Examples of Integrals
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Picture for the Definite Integral
Part of the Example on the Left
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| Look at the picture at the right
and notice that the definite integral below does not directly represent an
area of a region. Why?
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| One More Indefinite Integral
Example
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One More
Definite Integral Example
Can you tell which graph below goes with which
integral above, a or b? |
| Average Value of a Function
The areas of the regions enclosed in blue below
are equal. If f > 0 over the interval being looked at then the
average value of f over the interval will equal the height of a rectangle
whose base is equal in length to the length of the interval. |
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| Second Fundamental Theorem of
Calculus
If f is continuous on an open interval I
containing a, then, for every x in the interval
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| Find a function
of x that gives the area of the region shown in the picture at the
right. The curve forming the top border is part of the graph of y =
x2. The other borders are two vertical line segments and
part of the x-axis. Click here
or on the picture to see an animation corresponding to changing values of
x.
Animation
with shading and Quicktime version
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| Second Fundamental Theorem of
Calculus and the Chain Rule
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Total distance traveled when the position function
is non-decreasing.
Thus if F is a position function and f is its velocity
function, the total distance traveled for non-decreasing F would be
Observe that in the example below the position function
is not non-decreasing everywhere.
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Section 4.4 #97
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The
graph below shows position (not distance traveled) as a function of
time. Click here or on
the picture to see a linear motion animation. Quicktime
animation
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| Section 4.5 #121
You fill in the rest of the details to answer the
questions posed in the text. In the pictures, f(x) is graphed in red,
g(x) is graphed in green,
and h(x) is graphed in blue.
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Simpson's
Rule and the Trapezoidal Rule
You will need
to be able to describe in geometric terms what is going on when Simpson's Rule
is applied and when the Trapezoidal Rule is applied. Here are two
animations to assist you. The animations relate to using the Trapezoidal
Rule and using Simpson's Rule to compute the area of the region bounded by the
graphs of the given equations.
| P. 307 Review Exercises #13
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| P. 311 P.S. #7
In the picture at the right the parabola is y = 9
- x2 so a = 9. You fill in the details to the questions
asked in the text.
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