Area of Regions Between Curves

Area of Regions Between Curves

Area of a Region Between Two Curves (Integrate with Respect to x)

  • Definition: Area between two curves when integrating with respect to x.

    • If f(x) > g(x) on the interval [a, b], the area A of the region bounded by the graphs of f and g on [a, b] is given by:

    A = \int_{a}^{b} (f(x) - g(x)) dx

    • Case 1: If g(x) = 0, then:

    A = \int_{a}^{b} f(x) dx

    • Case 2: If g(x) \neq 0, then:

    A = \int_{a}^{b} (f(x) - g(x)) dx

Examples

  • Rectangle:

    • To find the area of a rectangle, integrate f(x) = h from 0 to l.

    A = A=\int_0^{l}\!f\left(x\right)\,dx=\int_0^{l}\!h\,dx

  • Triangular Plate:

    • To find the area of a triangular plate, integrate from -1 to 1.

    A=\int_{-1}^1\!f\left(x\right)\,dx

    =\int_{-1}^0\left(1+x\right)\!\,dx+\int_0^1\!\left(1-x\right)\,dx

    =\left\lbrack x+\frac{x^2}{2}\right\rbrack-1^0+\left\lbrack x-\frac{x^2}{2}\right\rbrack0^1=1

  • Area Between Two Curves:

    • Find the area of the region bounded by f(x) = 5 - x^2 and g(x) = x^2 - 3.

      1. Find Intersections:

      f(x) = g(x) \Rightarrow 5 - x^2 = x^2 - 3 \Rightarrow 2x^2 = 8 \Rightarrow x^2 = 4 \Rightarrow x = -2, 2

      1. Calculate Area:

      A = \int{-2}^{2} (5 - x^2) - (x^2 - 3) dx = \int{-2}^{2} (8 - 2x^2) dx = 2 \int_{0}^{2} (8 - 2x^2) dx

      *Because the function is even.*

      = 2 \Big[8x - \frac{2}{3}x^3\Big]_{0}^{2} = \frac{64}{3}

  • Area in the First Quadrant:

    • Find the area of the region in the first quadrant bounded by f(x) = x^{\frac{2}{3}} and g(x) = x - 4.

      1. Find Intersection:

      x^{\frac{2}{3}} = x - 4 \Rightarrow x^2 = (x - 4)^3 \Rightarrow x = 8

      *The intersection point is (8, 4).*

      1. Calculate Area:

      R = \int{0}^{4} f(x) dx + \int{4}^{8} f(x) - g(x) dx = \int{0}^{4} x^{\frac{2}{3}} dx + \int{4}^{8} x^{\frac{2}{3}} - (x - 4) dx

      = \Big[\frac{3}{5}x^{\frac{5}{3}}\Big]{0}^{4} + \Big[\frac{3}{5}x^{\frac{5}{3}} - \frac{1}{2}x^2 + 4x\Big]{4}^{8} = \frac{56}{5}

Area of a Region Between Two Curves (Integrate with Respect to y)

  • Definition: Area between two curves when integrating with respect to y.

    • If f(y) \geq g(y) on the interval [c, d], the area A of the region bounded by the graphs x = f(y) and x = g(y) on [c, d] is given by:

    A = \int_{c}^{d} (f(y) - g(y)) dy

Example (Revisited)

  • Area in the First Quadrant (Integrating with respect to y):

    • Find the area of the region in the first quadrant bounded by f(x) = x^{\frac{2}{3}} and g(x) = x - 4 by integrating with respect to y from [c, d] = [0, 4].

      1. Rewrite Functions in terms of y:

      f(x) = x^{\frac{2}{3}} \Rightarrow y = x^{\frac{2}{3}} \Rightarrow y^{\frac{3}{2}} = x \Rightarrow f(y) = y^{\frac{3}{2}}

      g(x) = x - 4 \Rightarrow y = x - 4 \Rightarrow y + 4 = x \Rightarrow g(y) = y + 4

      1. Calculate Area:

      R = \int{0}^{4} g(y) - f(y) dy = \int{0}^{4} (y + 4) - y^{\frac{3}{2}} dy

      = \Big[\frac{y^2}{2} + 4y - \frac{2}{5}y^{\frac{5}{2}}\Big]_{0}^{4} = \frac{56}{5}