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Aim: How do we find the area of a region between two curves?

Aim: How do we find the area of a region between two curves?. Do Now:. Area of Region Between 2 Curves. f ( x ). g ( x ). Area of region between f ( x ) and g ( x ). Representative Rectangle. area of representative rectangle.

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Aim: How do we find the area of a region between two curves?

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  1. Aim: How do we find the area of a region between two curves? Do Now:

  2. Area of Region Between 2 Curves f(x) g(x) Area of region between f(x) and g(x)

  3. Representative Rectangle area of representative rectangle

  4. If f and g are continuous on [a, b] and g(x) <f(x) for all x in [a, b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is Area of Region Between 2 Curves True regardless of relative position of x-axis; as long as f and g are continuous.

  5. Model Problem Find the area of the region bounded by the graphs of y = x2 + 2, y = -x, x = 0, and x = 1. Check with Calculator

  6. points of intersection Model Problem Find the area of the region bounded by the graph of f(x) = 2 – x2 and g(x) = x. a and b ?

  7. points of intersection a and b? Model Problem The sine and cosine curves intersect infinitely many times, bounding regions of equal areas. Find the area of one of these regions.

  8. Aim: How do we find the area of a region between two curves? Do Now: Find the area of the region between the graphs of f(x) = 1 – x2 and g(x) = 1 – x.

  9. points of intersection Model Problems Find the area of the region between the graphs of f(x) = 3x3 – x2 – 10x and g(x) = -x2 + 2x.

  10. Model Problems Find the area of the region between the graphs of f(x) = 3x3 – x2 – 10x and g(x) = -x2 + 2x.

  11. vertical rectangle horizontal rectangle Horizontal Representative Rectangles (Slices) Problem No Problem If a region is bounded by f(y) on the right and g(y) on the left at all points of the interval [c, d], then the area of the region is given by integrate with respect to y

  12. Horizontal Representative Rectangles If the graph of a function of y is a boundary of a region, it is often convenient to use representative rectangles that are horizontal and find the area by integrating with respect to y.

  13. left boundary: x = y2 right boundary: x = y + 6 Model Problem Find the area of the region between the curve x = y2 and the curve x = y + 6 from y = 0 to y = 3. for entire region:

  14. points of intersection Representative Rectangle Find the area of the region bounded by the graphs of x = 3 – y2 and x = y + 1. f(y) is to the right of g(y) (2, 1) Δy (-1, -2) area of representative rectangle

  15. points of intersection Representative Rectangle Find the area of the region bounded by the graphs of x = 3 – y2 and x = y + 1. f(y) is to the right of g(y) (2, 1) Δy (-1, -2)

  16. points of intersection Representative Rectangle Find the area of the region bounded by the graphs of x = 3 – y2 and x = y + 1. y = x – 1 Solve for y (2, 1) y = x – 1 (3, 0) (-1, -2) Δx x-intercept – (3, 0)

  17. Model Problem Find the area of the region between the curve y = sin x and the curve y = cos x from 0 to /2.

  18. Model Problem Find the area of the region between the curve

  19. Model Problem Find the area of the region between the curve and the x-axis from x = -3 to x = 3.

  20. Model Problem

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