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Estimating Areas Under Curves Using Finite Sums and Rectangular Approximation Methods

In this review, you will learn about estimating the area under curves using finite sums, laying a crucial foundation for integral calculus. We will explore methods such as the Left-hand Rectangular Approximation Method (LRAM), Right-hand Rectangular Approximation Method (RRAM), and Midpoint Rectangular Approximation Method (MRAM). Understanding these techniques is essential for calculating distance traveled by objects moving at varying rates. With increased subintervals, accuracy improves, making these methods valuable, even when the exact equation of the curve isn't available.

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Estimating Areas Under Curves Using Finite Sums and Rectangular Approximation Methods

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  1. 5.1 Estimating with Finite Sums

  2. Quick Review

  3. Quick Review Solutions

  4. What you’ll learn about • Distance Traveled • Rectangular Approximation Method (RAM) … and why Learning about estimating with finite sums sets the foundation for understanding integral calculus.

  5. velocity time Consider an object moving at a constant rate of 3 ft/sec. Since rate . time = distance: If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line. After 4 seconds, the object has gone 12 feet.

  6. Approximate area: If the velocity is not constant, we might guess that the distance traveled is still equal to the area under the curve. (The units work out.) Example: We could estimate the area under the curve by drawing rectangles touching at their left corners. This is called the Left-hand Rectangular Approximation Method (LRAM).

  7. Approximate area: We could also use a Right-hand Rectangular Approximation Method (RRAM).

  8. Approximate area: Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM). In this example there are four subintervals. As the number of subintervals increases, so does the accuracy.

  9. Approximate area: The exact answer for this problem is . With 8 subintervals: width of subinterval

  10. Circumscribed rectangles are all above the curve: Inscribed rectangles are all below the curve:

  11. We will be learning how to find the exact area under a curve if we have the equation for the curve. Rectangular approximation methods are still useful for finding the area under a curve if we do not have the equation. The graphing calculators can do these rectangular approximation problems with the correct programs. I can give you these in class. This is of limited usefulness, however, since we will learn better methods of finding the area under a curve, but you could use the calculator to check your work.

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