AP Calculus AB: 9.4.2 Areas, Riemann Sums, and Definite Integrals
This section explains how Riemann sums are used to approximate the area under a curve and how taking the limit of these sums leads to the definite integral, which gives the exact area. It connects the concepts of function, area, and antiderivative, showing how the area under a graph is computed through integration as the width of rectangles approaches zero.
Areas, Riemann Sums, and Definite Integrals
As the number of rectangles used to approximate the area of a region increases, the approximation becomes more accurate. It is possible to find the exact area by letting the width of each rectangle approach zero, thus generating an infinite number of rectangles.
The Riemann sum of f for the partition Δ is the sum , where xi – 1 ≤ ci ≤ xi , f is defined on [a, b], Δ is a partition of [a, b] given by a = x0 < x1 < … < xn – 1 < xn = b, and Δxi is the length of the with subinterval.
A function and the equation for the area between its graph and the x-axis are related by the antiderivative.
The definite integral of f from a to b is the limit of the Riemann sum as the lengths of the subintervals approach zero.
Key Terms
Areas, Riemann Sums, and Definite Integrals
As the number of rectangles used to approximate the area of a region increases, the approximation becomes more accurate. It is possible to ...
note
One way to approximate the area of a region is to fill it with rectangles. The sum of their areas will be an approximation for the area of ...
How is the concept of the limit important in finding the area of exotic shapes?
By taking the limit as Δx approaches zero, you can integrate the height of the shape along the width.
Use a left Riemann sum with three subintervals of equal length to approximate the area between the x‑axis and the graph of f (x) = 7 + 6x − x^2 for 0 ≤ x ≤ 3. That is, find the sum of the area of the three rectangles shown in the figure.
34
Use a right Riemann sum with three subintervals of equal length to approximate the area between the x‑axis and the graph of f (x) = 7 + 6x − x^2 for 0 ≤ x ≤ 3. That is, find the sum of the area of the three rectangles shown in the figure.
43
What is wrong with the labeling of this area problem?
The dx and f (x) are switched
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| Term | Definition |
|---|---|
Areas, Riemann Sums, and Definite Integrals |
|
note |
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How is the concept of the limit important in finding the area of exotic shapes? | By taking the limit as Δx approaches zero, you can integrate the height of the shape along the width. |
Use a left Riemann sum with three subintervals of equal length to approximate the area between the x‑axis and the graph of f (x) = 7 + 6x − x^2 for 0 ≤ x ≤ 3. That is, find the sum of the area of the three rectangles shown in the figure. | 34 |
Use a right Riemann sum with three subintervals of equal length to approximate the area between the x‑axis and the graph of f (x) = 7 + 6x − x^2 for 0 ≤ x ≤ 3. That is, find the sum of the area of the three rectangles shown in the figure. | 43 |
What is wrong with the labeling of this area problem? | The dx and f (x) are switched |
Use a left Riemann sum with four subintervals of equal length to approximate the area between the x‑axis and the graph of f (x) = x^2 + 2 for 0 ≤ x ≤ 2. That is, find the sum of the area of the four rectangles shown in the figure. | 5.75 |
Why does the definite integral of f (x) dx evaluated from a to b equal the area bound between a curve and the x‑axis? | Integrating the height of the function with respect to the width sums up all the tiny areas created by dividing the area into an infinite number of rectangles. |
Which of the following illustrations is properly | The properly labeled illustration shows a curve f(x)f(x)f(x) with vertical boundaries at x=ax = ax=a and x=bx = bx=b, shading the area under the curve between these points. |
Which of the following illustrations is | The properly labeled illustration shows a curve f(y)f(y)f(y) with horizontal boundaries at y=ay = ay=a and y=by = by=b, shading the area to the left of the curve between these values. |
What is wrong with the labeling of this area problem? | The b and the f (b) are switched. |
Use a right Riemann sum with four subintervals of equal length to approximate the area between the x‑axis and the graph of f (x) = x2 + 2 for 0 ≤ x ≤ 2. That is, find the sum of the area of the four rectangles shown in the figure. | 7.75 |