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The Area Between Two Curves

The Area Between Two Curves. Lesson 6.1. What If … ?. We want to find the area between f(x) and g(x) ? Any ideas?. When f(x) < 0. Consider taking the definite integral for the function shown below. The integral gives a negative area (!?) We need to think of this in a different way. a. b.

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The Area Between Two Curves

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  1. The Area Between Two Curves Lesson 6.1

  2. What If … ? • We want to find the area betweenf(x) and g(x) ? • Any ideas?

  3. When f(x) < 0 • Consider taking the definite integral for the function shown below. • The integral gives a negative area (!?) • We need to think of this in a different way a b f(x)

  4. Another Problem • What about the area between the curve and the x-axis for y = x3 • What do you get forthe integral? • Since this makes no sense – we need another way to look at it Recall our look at odd functions on the interval [-a, a]

  5. We take the absolute value for the interval which would give us a negative area. Solution • We can use one of the properties of integrals • We will integrate separately for -2 < x < 0 and 0 < x < 2

  6. General Solution • When determining the area between a function and the x-axis • Graph the function first • Note the zeros of the function • Split the function into portions where f(x) > 0 and f(x) < 0 • Where f(x) < 0, take absolute value of the definite integral

  7. Try This! • Find the area between the function h(x)=x2 + x – 6 and the x-axis • Note that we are not given the limits of integration • We must determine zeros to find limits • Also must take absolutevalue of the integral sincespecified interval has f(x) < 0

  8. Area Between Two Curves • Consider the region betweenf(x) = x2 – 4 and g(x) = 8 – 2x2 • Must graph to determine limits • Now consider function insideintegral • Height of a slice is g(x) – f(x) • So the integral is

  9. The Area of a Shark Fin • Consider the region enclosed by • Again, we must split the region into two parts • 0 < x < 1 and 1 < x < 9

  10. Slicing the Shark the Other Way • We could make these graphs as functions of y • Now each slice isy by (k(y) – j(y))

  11. Practice • Determine the region bounded between the given curves • Find the area of the region

  12. Horizontal Slices • Given these two equations, determine the area of the region bounded by the two curves • Note they are x in terms of y

  13. Assignments A • Lesson 7.1A • Page 452 • Exercises 1 – 45 EOO

  14. Integration as an Accumulation Process • Consider the area under the curve y = sin x • Think of integrating as an accumulation of the areas of the rectangles from 0 to b b

  15. Integration as an Accumulation Process • We can think of this as a function of b • This gives us the accumulated area under the curve on the interval [0, b]

  16. Try It Out • Find the accumulation function for • Evaluate • F(0) • F(4) • F(6)

  17. Applications • The surface of a machine part is the region between the graphs of y1 = |x| and y2 = 0.08x2 +k • Determine the value for k if the two functions are tangent to one another • Find the area of the surface of the machine part

  18. Assignments B • Lesson 7.1B • Page 453 • Exercises 57 – 65 odd, 85, 88

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