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Approximations of AreaStudent 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.

National Math and Science Initiative 2012


Let's try one together…

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.

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) Left Riemann Sum Right Riemann SumIncreasing (f '(x) > 0) Under approximates the Over approximates the



area because f(x) is increasing

area because f(x) is increasing

Decreasing (f '(x) < 0)Over approximates the

area because f(x) is decreasing

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.

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.

A=h2 (b1+b2 )

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 ApproximationsDetermining 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) Trapezoidal SumConcave Up (f ''(x) > 0) Over approximates the

area because f ''(x) > 0Concave Down (f ''(x) < 0) Under approximates the

area because f ''(x) < 0Multiple Choice Problems (From various released exams)

1. (Calculator active) Calculate the approximate area of the shaded region in the figure (below right) by the trapezoidal rule, using divisions at x= 4

3∧x=5

3.

A. 1.852B. 2.324C. 2.333D. 2.352E. 2.852

2. (Calculator active) If the definite integral ∫0

2

ex2

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. 53.60B. 30.51C. 27.80



D. 26.80E. 12.78

3. (Calculator active) A table of values for a continuous function f is shown below. If four equal subintervals of [0, 2] are used, which of the following is the trapezoidal approximation of ∫

0

2

f ( x )dx?

A. 8B. 12C. 16D. 24E. 32

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. 500B. 600C. 2,400


x 0 0.5 1.0 1.5 2.0f(x) 3 3 5 8 13


D. 3,000E. 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 approximation of ∫

2

8

f ( x )dx?

A. 110B. 130C. 160D. 190E. 210

6. (Calculator active) If a trapezoidal sum overapproximates ∫0

4

f ( x )dx , and a right

Riemann Sum underapproximates ∫0

4

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)

From the 2007 AP Calculus AB Exam (No calculators)



From the 2012 AP Calculus AB Exam (Calculator Active)



From the 2009 AP Calculus AB Exam (No calculators)


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· Web viewYou 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