Approximations of Area
Student Saturday Session
Approximating techniques are an important part of the AP Calculus exam. Earlier in the year, you
learned how to approximate the instantaneous rate of change at a point (derivative) of a function by
finding the slope between two points near the desired point.
You also learned to approximate values of a function near a specific point by using the tangent line
through that point (tangent line approximation)
Now, we will explore the process of approximating a definite integral (area between two curves). This is
an important concept and is usually tested multiple times on the AP Calculus exam.
When approximating the area under a curve on the AP exam, you will be asked to either use trapezoids as
your shape to approximate the area or rectangles (Riemann Sums). If you are asked to find a Riemann
Sum, this implies the use of a rectangle and you will need to determine the height and width of each
rectangle. The height of each rectangle depends on whether you are asked to find a Left, Right or
Midpoint Riemann Sum. Below, you can see graphically how the left, right or midpoint of each rectangle
can determine its height.
Left, Right and Midpoint Riemann Sums
If given an equation, f(x), you find the heights of each rectangle by plugging in appropriate x-values to
find the height. However, it is highly unlikely that you will be asked to find an approximation of area
when given an equation of a function. Why do you think this is?
You are much more likely to be asked to approximate area when given a table of values. When
approximating area using a Riemann Sum or Trapezoids, pay careful attention to the values you input and
the desired number of subintervals.
Let's try one together…
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Approximations of Area
Student Saturday Session
From the 2012 AP Calculus AB Practice Exam:
Two important notes from this problem:
1. We were able to approximate the total number of liters of oil that entered from t = 4 to t = 15. We still
needed to add the original amount to find the total liters in the tank at t = 15.
2. The rectangles did NOT have equal bases. This is usually true when dealing with Riemann Sums and
Trapezoidal Approximations. Pay close attention to the spacing along the x-axis.
Over and Under Approximations
If a function is strictly increasing or decreasing over an interval, then we are able to determine whether a
left/right Riemann Sum is an over or under approximation.
This is a common question on the Free Response Portion of the AP Calculus Exam that students often
miss due to improper explanations.
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Approximations of Area
Student Saturday Session
Let's consider the following problem:
Let f(x) be a function with f '(x) > 0 for all x-values in its domain. A sketch of the function and a left
Riemann Sum approximation is below. Is the Left Riemann Sum an over or under approximation?
Explain your reasoning.
Most students do a good job of writing that the left Riemann Sum is an under approximation. However,
when explaining their reasoning they often fail to give mathematical support and earn no points.
Incorrect Reasoning: The left Riemann Sum is an under approximation because the rectangles are all
underneath or below the graph.
Stating that the rectangles are below the function is not acceptable mathematical reasoning. It merely
restates that it is an under approximation but does not explain WHY.
What do you think would be acceptable as the reasoning for your answer?
Correct Reasoning:
f(x)
Increasing (f '(x) > 0)
Decreasing (f '(x) < 0)
Left Riemann Sum
Under approximates the area
because f(x) is increasing
Over approximates the area
because f(x) is decreasing
Right Riemann Sum
Over approximates the area
because f(x) is increasing
Under approximates the area
because f(x) is decreasing
Connection: On the AP Calculus exam, students are often asked to determine whether over/under
approximations occur with both Riemann Sums and Tangent Line Approximations.
Remember that Riemann Sums use f '(x) (f(x) increasing/decreasing) to make this determination
but Tangent Line Approximations use f ''(x) (f(x) concave up/down) to make the determination.
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Approximations of Area
Student Saturday Session
Trapezoidal Approximations
Another approximation technique is to use trapezoids to approximate the area instead of rectangles. In
order to find an approximate area using trapezoids, you will need to use the formula for the area of a
trapezoid.
ℎ
𝐴 = (𝑏1 + 𝑏2 )
2
Recall: The bases of a trapezoid are the two sides that are PARALLEL and the height is the altitude
between the two bases.
Note: A trapezoidal sum is the average of the left and right Riemann Sums---Can you see why? Since it
averages these two Riemann Sums, it tends to be more accurate than a Riemann Sum.
On these problems, the bases of the trapezoid will always run vertically and the height will be the distance
between the bases on the x-axis. Further, the bases are determined by the function you are using to
approximate area. The y-value of the function is the length of the base at a given x-value.
Textbooks will sometimes give a general formula to approximate area using trapezoids, but these
formulas require that each height is the same on the trapezoids. This is almost certainly NOT the case on
the AP Exam. Expect to be asked to find a trapezoidal approximation on a problem that has tabular data
given and count on the spacing to vary for each trapezoid. You will need to calculate the area of each
trapezoid individually (similar to Riemann Sums). Thankfully, you will not be asked to calculate more
than 3 or 4 trapezoids for your sum.
Over/Under Approximations with Trapezoidal Approximations
Determining whether a trapezoidal approximation is an over/under approximation of the true area under
the curve requires different logic and justifications that a Riemann Sum. Notice, in the illustration above,
the each trapezoid drawn is an underapproximation, even though the function increases and decreases
over the given interval.
This is because we determine whether trapezoidal approximations are over/under approximations from the
CONCAVITY of the function.
f(x)
Concave Up (f ''(x) > 0)
Concave Down (f ''(x) < 0)
Trapezoidal Sum
Over approximates the area
because f ''(x) > 0
Under approximates the area
because f ''(x) < 0
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Approximations of Area
Student Saturday Session
Multiple Choice Problems (From various released exams)
1. (Calculator active) Calculate the approximate area of the shaded region in the figure (below right) by
4
5
the trapezoidal rule, using divisions at 𝑥 = 3 𝑎𝑛𝑑 𝑥 = 3.
A. 1.852
B. 2.324
C. 2.333
D. 2.352
E. 2.852
2
2
2. (Calculator active) If the definite integral ∫0 𝑒 𝑥 𝑑𝑥 is first approximated by using a Left Riemann Sum
with two subintervals of equal width and then approximated by using the trapezoidal rule with two equal
subintervals, the difference between the two approximations is:
A. 53.60
B. 30.51
C. 27.80
D. 26.80
E. 12.78
3. (Calculator active) A table of values for a continuous function f is shown below. If four equal
2
subintervals of [0, 2] are used, which of the following is the trapezoidal approximation of ∫0 𝑓(𝑥)𝑑𝑥?
A. 8
B. 12
x
f(x)
0
3
0.5
3
1.0
5
1.5
8
C. 16
D. 24
E. 32
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Approximations of Area
Student Saturday Session
4. (No calculator) The flow of oil, in barrels per hour, through a pipeline on July 9 is given by the graph
shown above. Of the following, which best approximates the total number of barrels of oil passed
through the pipeline that day?
A. 500
B. 600
C. 2,400
D. 3,000
E. 4,800
5. (Calculator Active) The function f is continuous on the closed interval [2,8] and has values that are
given in the table above. Using the subintervals [2,5], [5,7], and [7,8], what is the trapezoidal
8
approximation of ∫2 𝑓(𝑥)𝑑𝑥?
A. 110
B. 130
C. 160
D. 190
E. 210
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Approximations of Area
Student Saturday Session
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6. (Calculator active) If a trapezoidal sum overapproximates ∫0 𝑓(𝑥)𝑑𝑥, and a right Riemann Sum
4
underapproximates ∫0 𝑓(𝑥)𝑑𝑥, which of the following could be a graph of y = f(x)?
Free Response Questions
From the 2010 AP Calculus AB Exam (Calculator Active)
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Approximations of Area
Student Saturday Session
From the 2007 AP Calculus AB Exam (No calculators)
National Math and Science Initiative 2012
Approximations of Area
Student Saturday Session
From the 2012 AP Calculus AB Exam (Calculator Active)
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Approximations of Area
Student Saturday Session
From the 2009 AP Calculus AB Exam (No calculators)
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