advertisement

Math 152, Fall 2008 Lecture 2. 08/28/2008 Example 1. Evaluate each integral. (a) Z4 1 x2 r 1 1 + dx x (b) e 1 (c) Z sin x dx 1 + cos2 x Zπ/2 (e) x 2 sin x dx 1 + x6 −π/2 (j) Z Ze 4 arcsin x + x √ dx 1 − x2 (d) (f) R Z dx √ x ln x t 2 cos(1 − t 3 )dt x2 dx + a2 Chapter 7. Applications of integration Section 7.1 Areas between curves Consider the region S that lies between two curves y = f (x) and y = g (x) and between the vertical lines x = a and x = b, where f and g are continuous functions and f (x) ≥ g (x) for all x in [a, b]. y y = f (x) S 0 a b x y = g(x) S = {(x, y ) : a ≤ x ≤ b, g (x) ≤ y ≤ f (x)} Let P be a partition of [a, b] by points xi and choose points xi∗ in [xi −1 , xi ]. Let ∆xi = xi − xi −1 and kPk = max{∆xi }. n P The Riemann sum [f (xi∗ ) − g (xi∗ )]∆xi is an approximation to i =1 what we intuitively think as the area of S. This approximation appears to become better and better as kPk → 0. We define the area A of S as n P [f (xi∗ ) − g (xi∗ )]∆xi Therefore: A = lim kPk→0 i =1 The area of the region bounded by the curves y = f (x), y = g (x), and the lines x = a and x = b, where f and g are continuous functions and f (x) ≥ g (x) for all x in [a, b], is Rb A = [f (x) − g (x)]dx a Example 2. Find the area of the region bounded by the parabolas y = 4x 2 , y = x 2 + 3 In general case, the area between the curves y = f (x), y = g (x) Rb and between x = a and x = b, is A = |f (x) − g (x)|dx a Example 3. Find the area of the region bounded by the curves y = cos x, y = sin 2x an the lines x = 0 and x = π/2. Example 4. Find the area of the shaded region. y y = x x = y2 − 2 0 x If a region is bounded by curves with equations x = f (y ), x = g (y ), y = c and y = d, where f and g are continuous functions and f (y ) ≥ g (y ) for all y in [c, d], then its area is Rd A = [f (y ) − g (y )]dy c y d S x = f (y) x = g(y) c 0 x