Homework: Study 6.1 # 1, 5, 7, 13, 25, 19; 3, 17, 27, 53

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1 January, 7 Goals:. Remember that the area under a curve is the sum of the areas of an infinite number of rectangles. Understand the approach to finding the area between curves.. Be able to identify the region bounded by the given conditions. 4. Determine the best orientation for the reference rectangle in the region horizontal or vertical. 5. Set up the definite integral for finding the area. Homework: Study 6. #, 5, 7,, 5, 9;, 7, 7, 5 area between curves To find the area under the curve over a specified interval, we would need to divide it into lots of little polygons, such as rectangles, for which we can find the area with known formulas. Background: Area Under a Curve y = x(x 4) where n is the number of rectangles Areas of small rectangles 7

2 Background: Area Under a Curve Definition of Area of a Region in a Plane Let f be continuous and non negative on [a,b]. The area of the region bounded by the graph of f, the x axis, and the vertical lines x = a and x = b is: January, 7 x i < c i < x i Δx = b a n as n >, Δx > F: Area of shaded region A of rectangle = g(c i )Δx b f(x)dx = F(b) F(a) a 7

3 January, 7 A = (4 x ) dx = 4x x = 4() 8 = 8 8 = 6 F: Area of shaded region A of rectangle = f(c i )Δx 7

4 January, 7 A = x dx = x = 8 = 8 How to find Area of shaded region? st need to find point of intersection, because region does not extend beyond that point. 7 4

5 January, 7 How to find Area of shaded region? st need to find point of intersection, because region does not extend beyond that point. x = 4 x x = 4 x = x = ± UL, x = + LL, x = How to find Area of shaded region? A C area betw curves B D C shows both A and B together D shows A B 7 5

6 January, 7 How to find Area of shaded region? How do we do that analytically? Use reference rectangle. A of rectangle = [g(c i ) f(c i )]Δx Δx g(c i ) f(c i ) A = (4 x x ) dx (4 x ) dx = 4x x = 4 4 = 8 Area of region Bounded by Curves If f and g are continuous on [a, b] and g(x) < f(x) for all x on [a, b], then the area of the region bounded by f and g and the vertical lines and x = a and x = b is: A = a b [ f(x) g(x) ] dx In general, the area between curves that intersect and switch y positions is given by: A = a b f(x) g(x) dx for vertical reference rectangle or A = c d f(y) g(y) dy for horizontal reference rectangle 7 6

7 January, 7 G: f(x) = (x ), g(x) = x f(x) = (x ) g(x) = x G: f(x) = (x ), g(x) = x f(x) = (x ) A = (f g)dx + (g f)dx A =½ (x )(x x)dx + ½ (x )(x x)dx u= x x du=(x )dx=(x )dx =½(x x) ½(x x) =¼( ) [¼(4 4) ¼( ) ] = +¼ [ ¼ ] = ¼ +¼ = ½ check on calculator: b g(x) = x A= f g dx= abs[(x ) (x )]dx =.5 a Δx f g Δx g f point of intersection: (x ) = x (x ) (x )= (x ) [(x ) ]= (x ) [(x x+ ]= (x ) (x x)= (x )(x)(x )= x=,, 7 7

8 January, 7 G: f(x) = x, h(x) =, x= x Axis To use a vertical reference rectangle, need to break up the region into parts. area between curves G: f(x) = x, h(x) =, x= x Axis Use a horizontal reference rectangle instead. Then can do the whole region at once, using dy. Need x from f(x): y = x so x = y x = y ( y ) dy = y y = / = 5/ Δy area between curves 7 8

9 January, 7 G: f(x) = (x ), g(x) = x f(x) = (x ) x f x g x g x f Δy Δy g(x) = x area between curves G: f(x) = (x ), g(x) = x A= [(y ) ( y + )]dy + y = x f x f = y + = y y 4/ + y 4/ y 4/ 4/ x g = y [( y +) (y )]dy A= (y y / )dy + (y / y+)dy x g x f Δy g(x) = x f(x) = (x ) x f x g Δy = [/ /4] + [/4 / ] = / + /4 + /4 / = / = / 7 9

10 Summary of Finding Area betw Curves. Sketch the curves and the find points of Intersection. Use the sketch to determine which integral to use: ο If each curve passes the vertical line test, then Use a vertical rectangle, the x variable, and dx: A = (upper lower) dx ο If a curve fails the vertical line test but passes the horizontal line test, then use a horizontal rectangle, the y variable, and dy: A = (right left) dy. If the bounded area contains more than one distinct region, write the area as the sum of the areas of each distinct region. 4. Limits of Integration: ο Use the coordinates of the points of intersection. ο If x=k or y=k is given this may be one of the limits. January, 7 7