math152_7.1_7.4.pdf - Math 150B Jagodina 1 Chapter 7.1-7.4...

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Math 150B, Jagodina Chapter 7.1-7.4 Packet Tutors may help. 1 Area of a Region Between Curves Recall that we initially developed the definite integral to compute the area under a curve. Area = A = Z b a f ( x ) dx Suppose that we have two continuous functions, f and g , and f ( x ) g ( x ) for all x on [ a, b ]. We would like to find the area bounded by the graphs of y = f ( x ) and y = g ( x ) on the interval [ a, b ]. Formula: Area= Z b a f ( x ) - g ( x ) dx 1
Math 150B, Jagodina Chapter 7.1-7.4 Packet Tutors may help. Example Find the area bounded by the graphs of y = - x 2 and y = x + 1 for 0 x 2. Sketch the region. Example Find the area bounded by the graphs of y = 3 - x and y = x 2 - 9. Sketch the region. 2
Math 150B, Jagodina Chapter 7.1-7.4 Packet Tutors may help. Finding the Area of a Region Using Several Integrals Finding the area of some regions may require breaking the region up into several pieces, each having different upper and/or lower boundaries. Example Find the area bounded by the graphs of y = x 2 and y = 2 - x 2 for 0 x 2. Sketch the region. 3
Math 150B, Jagodina Chapter 7.1-7.4 Packet Tutors may help. Integrating with respect to y Formula: Area = A = Z d c f ( y ) - g ( y ) dy Example Find the area of the region bounded by the graphs of x = y 2 and x = 2 - y 2 . Sketch the region. 4
Math 150B, Jagodina Chapter 7.1-7.4 Packet Tutors may help. 2 Volume: The Disk Method If a region in the plane is revolved about a line, the resulting solid is a solid of revolution, and the line is called the axis of revolution. The simplest such solid is a right circular cylinder or disk, with is formed by revolving a rectangle about an axis adjacent to one side of the rectangle. The volume of a disk is V = πr 2 w = πr 2 dx or V = πr 2 dy If cross section is a disk, then Volume of a solid = V = Z b

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