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Area Connection and the FTC

This guide explores how to find the total area between the graph of ( y = f(x) ) and the x-axis over the interval ([a, b]). It covers two main methods: analytically, by partitioning the interval at the zeros of the function and integrating over each subinterval, then summing the absolute values of the integrals; and numerically, using NINT to evaluate the integral of the absolute value of the function over ([a, b]). Guided practice includes worked examples to reinforce understanding of these techniques.

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Area Connection and the FTC

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  1. Area Connectionand the FTC Section 5.4b

  2. How to Find Total Area Analytically To find the area between the graph of y = f (x) and the x-axis over the interval [a, b ] analytically, 1. Partition [a, b ] with the zeros of f, 2. Integrate f over each subinterval, 3. Add the absolute values of the integrals.

  3. How to Find Total Area Numerically To find the area between the graph of y = f (x ) and the x-axis over the interval [a, b ] numerically, evaluate NINT ( |f (x)|, x, a, b )

  4. Guided Practice Find the area of the region between the given function and the x-axis for the given interval. How about a graph? Over [0, 2]: 4 Antiderivative: 1 2 3 –5

  5. Guided Practice Find the area of the region between the given function and the x-axis for the given interval. How about a graph? Over [0, 2]: 4 1 2 3 –5

  6. Guided Practice Find the area of the region between the given function and the x-axis for the given interval. How about a graph? Over [2, 3]: 4 1 2 3 –5

  7. Guided Practice Find the area of the region between the given function and the x-axis for the given interval. How about a graph? Total area of the region: 4 1 2 3 Verify with NINT!!! –5

  8. Guided Practice Find the area of the region between the given function and the x-axis for the given interval. The graph? Total Area = 8

  9. Guided Practice For the following integral, (a) can the FTC (part 2) be used to evaluate the integral, and (b) does the integral have a value (if so, what is it? – explain!)? • The FTC (part 2) cannot be used, because the integrand is discontinuous at x = 3!!! Evaluate this integral graphically!!!

  10. Guided Practice For the following integral, (a) can the FTC (part 2) be used to evaluate the integral, and (b) does the integral have a value (if so, what is it? – explain!)? This integral does not have a value!!! Why not???

  11. Guided Practice For the following integral, (a) can the FTC (part 2) be used to evaluate the integral, and (b) does the integral have a value (if so, what is it? – explain!)? This integral does exist, because the region is bounded!!! (evaluate the integral numerically)

  12. Guided Practice Find the area of the shaded region for #26 on p.286. Total Area

  13. Guided Practice Find the area of the shaded region for #28 on p.286. Calculate the area under the sine curve over the interval, then subtract the area of the rectangle… Total Area

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