Introduction to US Healthcare Economics
This presentation delves into the economics of healthcare in the United States, exploring challenges, trends in healthcare expenditures, roles of major players such as consumers, providers, insurance, and government, and the impact of government interventions. Key questions, the overview of the US healthcare system, challenges faced, and national health expenditure trends are discussed.
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Volumes of Revolution LO: To calculate the volumes of revolution about the x-axis or the y-axis. www.mathssupport.org
Volumes of Revolution A solid of revolution is formed by rotating a plane figure about an axis of revolution First consider a rectangle perpendicular to the x-axis. Imagine rotating the rectangle 360o about the x-axis. y 0 x www.mathssupport.org
Volumes of Revolution A solid of revolution is formed by rotating a plane figure about an axis of revolution First consider a rectangle perpendicular to the x-axis. Imagine rotating the rectangle 360o about the x-axis. The solid form is called a disk. y 0 x www.mathssupport.org
Volumes of Revolution A solid of revolution is formed by rotating a plane figure about an axis of revolution First consider a rectangle perpendicular to the x-axis. Imagine rotating the rectangle 360o about the x-axis. The solid form is called a disk. The disk is cylindrical in shape. dx y Vcylinder. = r2h y Vdisk = y2dx 0 x Vdisk= (f(x))2dx www.mathssupport.org
Volumes of Revolution Consider the following solid of revolution formed when the region bounded by f(x) and the x-axis between x = a and x = b is rotated 360o about the x axis. www.mathssupport.org
Volumes of Revolution To calculate the volume of the solid we can divide it into slices by making cuts perpendicular to the axis of rotation. If the cuts are reasonably close together then each slice will approximately look like a cylinder. radius = y height = dx y dx We can then find volume of the solid by adding volumes of these small cylinders together. www.mathssupport.org
Volumes of Revolution We know that volume of a cylinder is: V = r2h y Hence, volume of the following cylinder with radius y and height dx is: V y2dx dx www.mathssupport.org
Volumes of Revolution The total volume of the solid is V, where: V y2dx Therefore, dx 0 = 2 lim V y dx y Which implies: = ``dx 2 V y dx www.mathssupport.org
Volumes of Revolution The volume V of a solid formed by a curve y = f(x) between x = a and x = b rotated through 2 radians about the x-axis is given by: ? ??2?? ? = ? ? ?2?? ? = ? ? www.mathssupport.org
Volumes of Revolution Use a definite integral to find the volume of the solid formed when the region bounded by f(x) = 9 ?2 and the x-axis is rotated 360o about the x-axis. Verify your answer using a geometric formula. Sketching the graph Drawing a representative rectangle is useful If we rotate the rectangle we will have a disk, radius = 9 ?2 The height of the disk is the width of the representative rectangle: dx 9 ?2The limits of integration are the x-intercepts, -3 and 3 y= 3 ? ??2?? ? = ? 3 -3 3 ?( 9 ?2)2??Use the GDC to dx ? = evaluate the integral 3 ? 113 www.mathssupport.org
Volumes of Revolution Use a definite integral to find the volume of the solid formed when the region bounded by f(x) = 9 ?2 and the x-axis is rotated 360o about the x-axis. Verify your answer using a geometric formula. When the region is rotated 360o about the x-axis a sphere is formed Volume of the sphere =4 3? ?3 9 ?2=4 3? (3)3 =36? y= 3 ? 113 -3 3 www.mathssupport.org
Volumes of Revolution Find the volume of the solid formed when the region under the curve f(x) = x2 between x = 0 and x = 5 is rotated 360o about the x-axis. Give the exact answer . Sketching the graph Rotating the graph The limits of integration are 0 and 5 ? ??2?? ? = y=?2 ? 5 ? ?2 2?? ? = 0 5 ?4?? ? = ? x 0 5 0 5 ?5 5 ? = ? 0 ? = ?(625 0) cubic units ? = 625? www.mathssupport.org
Volumes of Revolution The volume V of a solid formed by rotating a curve y = f(x) through 2 radians about the y-axis between y = c and y = d is given by: ? ??2?? ? = ? ? ?2?? ? = ? f(a) = c f(b) = d ? To use this formula directly you must first express x in terms of y www.mathssupport.org
Volumes of Revolution Find the volume of the solid formed by rotating the line f(x) = 2x between x = 0and x = 2 about the y-axis. The interval [0, 2] on the x-axis Corresponds to [0, 4] on the y-axis Rearrangingy=2? gives ? = Sketching the graph Rotating the graph ? 2 ? ?2?? ? = ? ? 4 4? 2 ? = ? ?? 2 y=2? 0 4 ? =? ?2?? 4 0 x 4 43 3 ?3 3 = ? =? 0 2 4 4 0 cubic units ? =16 3 www.mathssupport.org
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