AP Calculus AB: 9.4.1 Approximating Areas of Plane Regions
This section introduces the concept of approximating the area under a curve by dividing a region into rectangles, a foundational idea leading to definite integrals. It explains how increasing the number of rectangles improves accuracy and connects this approximation process to the use of limits and sigma notation in calculus. The method ultimately bridges the connection between geometry and integration.
Approximating Areas of Plane Regions
The two key questions of calculus have a subtle connection.
When trying to find the area of a complicated region, try approximating the area with rectangles. As the number of rectangles increases, the approximation becomes more accurate.
Key Terms
Approximating Areas of Plane Regions
The two key questions of calculus have a subtle connection.
When trying to find the area of a complicated region, try approx...
note
The two big questions in calculus are “How do you find
instantaneous velocity?” and “How do you find the area of exotic shapes?”
Approximate the area under the curve y = −x^ 2 + 16 between 0 and 4 by finding the area of the circumscribed rectangles using 4 subdivisions.
A ≈ 50
Which of the following approximations is closest to the actual area underneath this curve between a and b ?
The approximation closest to the actual area under the curve between a and b is the Trapezoidal Rule
When using rectangular partitions to approximate the area under a curve, how does increasing the number of partitions (or rectangles) used to approximate an area generally affect the approximation?
Increasing the number of partitions or rectangles generally generates a smaller error in the approximation.
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| Term | Definition |
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Approximating Areas of Plane Regions |
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Approximate the area under the curve y = −x^ 2 + 16 between 0 and 4 by finding the area of the circumscribed rectangles using 4 subdivisions. | A ≈ 50 |
Which of the following approximations is closest to the actual area underneath this curve between a and b ? | The approximation closest to the actual area under the curve between a and b is the Trapezoidal Rule if the function is approximately linear, or Simpson’s Rule if the function is smooth and curved. |
When using rectangular partitions to approximate the area under a curve, how does increasing the number of partitions (or rectangles) used to approximate an area generally affect the approximation? | Increasing the number of partitions or rectangles generally generates a smaller error in the approximation. |