1 / 15

Area between curves

Section 7.2a. Area between curves. Area Between Curves. Suppose we want to know the area of a region that is bounded above by one curve, y = f ( x ), and below by another, y = g ( x ):. Upper curve. First, we partition the region into vertical strips of equal

melody
Download Presentation

Area between 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. Section 7.2a Area between curves

  2. Area Between Curves Suppose we want to know the area of a region that is bounded above by one curve, y = f(x), and below by another, y = g(x): Upper curve First, we partition the region into vertical strips of equal width and approximate each strip as a rectangle with area a b Note: This expression will be non-negative even if the region lies below the x-axis. Lower curve

  3. Area Between Curves Suppose we want to know the area of a region that is bounded above by one curve, y = f(x), and below by another, y = g(x): Upper curve We can approximate the area of the region with the Riemann sum a b The limit of these sums as is Lower curve

  4. Definition: Area Between Curves If f and g are continuous with throughout [a, b], then the area between the curves y = f(x) and y = g(x) from a to b is the integral of [f – g] from a to b,

  5. Guided Practice Find the area of the region between and from to . Now, use our new formula to find the enclosed area: First, graph the two curves over the given interval:

  6. Guided Practice Find the area of the region between and from to . First, graph the two curves over the given interval:

  7. Guided Practice Find the area of the region enclosed by the parabola and the line . The graph: To find our limits of integration (a and b), we need to solve the system Algebraically, or by calculator: b a

  8. Guided Practice Find the area of the region enclosed by the parabola and the line . Because the parabola lies above the line, we have The graph: b a units squared

  9. Guided Practice Find the area of the region enclosed by the graphs of and . The graph: To find our limits of integration (a and b), we need to solve the system Solve graphically: b a Store the negative value as A and the positive value as B.

  10. Guided Practice Find the area of the region enclosed by the graphs of and . Note: The trigonometric function lies above the parabola… The graph: Let’s evaluate this one numerically… Area: b a units squared

  11. Guided Practice Find the area of the region R in the first quadrant that is bounded above by and below by the x-axis and the line . The graph of R: Area of region A: 2 (4,2) B 1 A 1 2 3 4

  12. Guided Practice The graph of R: Area of region B: 2 (4,2) B 1 A 1 2 3 4

  13. Guided Practice The graph of R: Area of R = Area of A + Area of B: 2 (4,2) B 1 A Units squared 1 2 3 4

  14. Guided Practice Find the area of the region enclosed by the given curves. First, graph in [–2,12] by [0,3.5] Intersection points: Break into three subregions:

  15. Guided Practice Find the area of the region enclosed by the given curves. First, graph in [–2,12] by [0,3.5] Intersection points: Break into three subregions:

More Related