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The Area Problem

The Area Problem. Lets take a trip back in time…to geometry . Can you find the area of the following? If so, why?. The Area Problem Continued.

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The Area Problem

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  1. The Area Problem Lets take a trip back in time…to geometry. Can you find the area of the following? If so, why?

  2. The Area Problem Continued Now, lets take a trip back to Advanced Algebra. Can you find the area of the region bounded by the line x=0, y=0 , y = 4 and y = 2x+3? If so, how? 3 4 0

  3. The Area Problem Continued One Last Time 16 4 0

  4. When we find the area under a curve by adding rectangles, the answer is called a Riemann Sum. The width of a rectangle is called a subinterval. The entire interval is called the partition. subinterval partition Subintervals do not all have to be the same size.

  5. If the partition is denoted by P, then the length of the longest subinterval is called the norm of P and is denoted by . As gets smaller, the approximation for the area gets better. Why? subinterval partition if P is a partition of the interval

  6. is called the definite integral of over . If we use subintervals of equal length, then the length of a subinterval is: The definite integral is then given by:

  7. Leibnitz introduced a simpler notation for the definite integral: Note that the very small change in x becomes dx.

  8. It is called a dummy variable because the answer does not depend on the variable chosen. upper limit of integration Integration Symbol integrand variable of integration (dummy variable) lower limit of integration

  9. We have the notation for integration, but we still need to learn how to evaluate the integral. This will be another day. We will master the Riemann Sum work first! Onwards!!!

  10. velocity time The Area Problem In Calculus Consider an object moving at a constant rate of 3 ft/sec. Since rate . time = distance: d’ ? 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. Why?

  11. Approx. 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. We call this the Left-hand Rectangular Approx. Method (LRAM).

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

  13. Approx 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.

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

  15. Circumscribed rectangles are all above the curve: Inscribed rectangles are all below the curve: What Riemann Method?Over or under estimate?Concave up or down? What Riemann Method?Over or under estimate?Concave up or down?

  16. Riemann Sums Exercise Handout

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