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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

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Area between curves

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  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:

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