University:
Kentucky State University
Course:
MAT 120 | Precalculus
Academic year:

2019
Views:

253

Pages:

2
Author:

teespagmaxrtf

**Area between two curves:** * Theorem: If f and g are continuous functions on the interval [a,b] such that f(x) ≥ g(x) for all x ∈ [a,b], then the area of the region bounded above by y = f(x), below by y = g(x), on the left by x = a, and on the right by x = b is: A = ∫[a,b] (f(x) - g(x)) dx **Example 1:** Find the area of the region enclosed by y = x², y = x + 6. x² = x + 6 x² - x - 6 = 0 (x - 3)(x + 2) = 0 x = 3 or x = -2 A = ∫[-2,3] (x + 6 - x²) dx **Example 2:** Find the area enclosed by y = e^x, y = e^-x, x = 0, x = ln2. e^x = e^-x e^(2x) - 1 = 0 (e^x - 1)(e^x + 1) = 0 e^x ≠ 0, so e^x - 1 = 0 => x = 0 A = ∫[0,ln2] (e^x - e^-x) dx **eg) Find the area enclosed by y = sinx, y = cosx, x = 0, x = π/2** sinx = cosx sinx/cosx = 1 tanx = 1 => x = π/4 A = ∫[0,π/4] (cosx - sinx) dx + ∫[π/4,π/2] (sinx - cosx) dx [Image of a graph with two shaded areas, one between 0 and π/4, and the other between π/4 and π/2] A₁ = ∫[0,π/4] (cosx - sinx) dx A₂ = ∫[π/4,π/2] (sinx - cosx) dx