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Calculus and Analytic Geometry I Cloud County Community College Fall, 2012 Instructor: Timothy L. Warkentin

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Chapter 06: Applications of Definite Integrals 6.1 Volumes Using Cross-Sections 6.2 Volumes Using Cylindrical Shells 6.3 Arc Length 6.4 Areas of Surfaces of Revolution 6.5 Work and Fluid Forces 6.6 Moments and Centers of Mass

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Chapter 06 Overview All the integral applications in this chapter are derived from the following statement presented in the previous chapter,, where c k is any x-value in the k th subinterval of [a,b], Δx = (b - a)/n, and F is any antiderivative of f [x].

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06.01: Volumes Using Cross-Sections 1 If the summation of f [c k ] Δx is to be volume then product f [c k ] Δx must itself be volume. The volume of any cylindrical solid is found by the product of (Area of the Base - B) and (height of the cylinder - h). Since f [c k ] Δx is a volume and Δx is a length then f [c k ] must be an area and Δx is the height of the cylinder. A formula f [x] for the base area of a cylinder at an arbitrary position x in [a,b] must be found and multiplied by Δx. If the sum of the volumes found in this way converges then the fundamental theorem can be applied to find the convergence value.

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06.01: Volumes Using Cross-Sections 2 Volumes by Parallel Slicing. Examples 1 & 2 Cavalieri’s Principle. Example 3 Volumes by Rotation: The Disk Method. Examples 4 – 8 Volumes by Rotation: The Washer Method. Examples 9 & 10

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06.02: Volumes Using Cylindrical Shells 1 The volume of a solid of revolution where the Washer Method would fail (because the inner and outer boundaries of the washer are given by the same curve) can be found by the Cylindrical Shells Method. Examples 1, 2 & 3 Summary of the Shell Method.

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06.03 Arc Length 1 Length of a curve where f is defined as a function of x. Examples 1 – 3 Length of a curve where g is defined as a function of y. Example 4

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06.03 Arc Length 2 Derivation of the short differential form for arc length. Example 5

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06.04 Areas of Surfaces of Revolution 1 Surface Area for Revolution About the x-Axis (f [x] ≥ 0). Example 1 Surface Area for Revolution About the y-Axis (g [y] ≥ 0). Example 2

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06.05: Work and Fluid Forces 1 The work done by a constant force is the product of the force and the distance the object has moved in the direction of the force. Example 1

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06.05: Work and Fluid Forces 2 Work done by a variable/constant force F[x] in the direction of motion from a to b is. The Riemann product is (entire force)(distance interval). Examples 1, 2, 3, 4 & 5, Spreadsheets, Area program Hooke’s Law for Springs: –F is the force needed to hold a spring x units (always a positive number) from its natural length of zero. –The work required is.

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06.05: Work and Fluid Forces 3 Mass Density:, Weight Density: The work done pumping fluids: The Riemann product is (force interval)(entire distance). Examples 4 & 5 Fluid Pressure is dependent only on the depth beneath the surface of the liquid. Pascal’s Principle: The pressure at a point in an incompressible liquid is the same in all directions. Pressure:

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06.05: Work and Fluid Forces 4 The fluid force against a submerged vertical plate: –The fluid force on a horizontal strip of length L[y] and width Δy across a vertical plate at a depth y:. Example 6

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06.06: Moments and Centers of Mass 1 Torque (the first moment or moment of force) is that which causes a body to rotate. Torque is the product of the perpendicular force applied to a point on a lever arm and the distance of that point from the point of rotation. The pivotal idea involved in finding the centers of mass is.

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06.06: Moments and Centers of Mass 2 Rewriting F i x i so that it is a Riemann product: F = m g = ρ V g = ρ g V = ρ g t h Δx. Example 1 The center of mass of simple shapes and symmetry. Using horizontal/vertical strips to find. Example 1 Using a density function to account for variable thickness. Examples 2 & 3 Pappus’s Theorems for volumes and surface areas. Examples 6 & 7

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Review: Volumes of Revolution. x y A 45 o wedge is cut from a cylinder of radius 3 as shown. Find the volume of the wedge. You could slice this wedge.

Review: Volumes of Revolution. x y A 45 o wedge is cut from a cylinder of radius 3 as shown. Find the volume of the wedge. You could slice this wedge.

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