Some Area Problems Area of -心fwa (x) 2 Figure 5.5.1 Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Some Area Problems Area of = b a f (x)dx
Some Area Problems Example 1 Find the area below the graph of the square-root function fromx=0 tox=1. y= Figure 5.5.2 Main Menu 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Some Area Problems Example 1 Find the area below the graph of the square-root function from x=0 to x=1
Some Area Problems Example 2 Fine the area of the region bounded above by the curve y=4-x2and below by the x-axis. y Figure 5.5.3 Main Meny 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Some Area Problems Example 2 Fine the area of the region bounded above by the curve y=4-x 2 and below by the x-axis
Some Area Problems y=f儿) y=f(r) y=g(x) area of Q=area of -area of 2 Figure 5.5.4 areaof=f(x)dk and areaof=[g(x)dx we have (5.5.2) area of [f(x)-g(x)]dx Main Menu o墙8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Some Area Problems area of and area of 1 2 ( ) ( ) b b a a = = f x dx g x dx we have
Some Area Problems Example 3 Find the area of the region bounded above by the line y=x+2 and bounded below by the parabolay=x2 Figure 5.5.5 Main Meny 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Some Area Problems Example 3 Find the area of the region bounded above by the line y = x + 2 and bounded below by the parabola y = x 2
Some Area Problems y y=f(x)+C y=f(x) 2' 2 y=g(x)+C 6 y=g(x) Figure 5.5.6 Figure 5.5.7 Main Meny 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Some Area Problems
Some Area Problems Example 4 Find the area of the region shown in Figure 5.5.8 Figure 5.5.8 Main Meny o8的
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Some Area Problems Example 4 Find the area of the region shown in Figure 5.5.8
Some Area Problems Example 5 Find the area between y=4x and y=x3 fromx=-2 tox=2 Figure 5.5.9 Main Meny 5
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Some Area Problems Example 5 Find the area between y = 4x and y = x 3 from x = -2 to x = 2
Some Area Problems Example 6 Use integrals to represent the area of the region 1=21U2 shaded in Figure 5.5.10 f(x) 2 2 Figure 5.5.10 Main Meny 007 e
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Some Area Problems Example 6 Use integrals to represent the area of the region Ω1 = Ω1∪Ω2 shaded in Figure 5.5.10
Some Area Problems 23 02 Figure 5.5.11 The area between the graph of fand the x-axis from x=a tox=e is the sum area of area of 2+area of ;area of This area is ∫f()k-∫f(x)+f(x)-∫if(x) Main Meny o
Main Menu Salas, Hille, Etgen Calculus: One and Several Variables Copyright 2007 © John Wiley & Sons, Inc. All rights reserved. Some Area Problems The area between the graph of f and the x-axis from x = a to x = e is the sum area of Ω1 + area of Ω2 + area of Ω3 + area of Ω4 This area is ( ) ( ) ( ) ( ) . b c d e a b c d f x dx f x dx f x dx f x dx − + −