1 / 18

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.

ismail
Download Presentation

The Area Between Two Curves

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related