Chapter 7 Applications of Integration

Transcription

1 Chapter 7 Applications of Integration 7.1 Area of a Region Between Two Curves 7.2 Volume: The Disk Method 7.3 Volume: The Shell Method 7.5 Work 7.6 Moments, Centers of Mass, and Centroids 7.7 Fluid Pressure and Fluid Force

2 Arc Length A rectifiable curve is one that has a finite arc length. A sufficient condition for the graph of a function f to be rectifiable between (a, f a ) and (b, f b ) is that f be continuous on [a, b]. Such a function is continuous differentiable on [a, b], and its graph on the interval [a, b] is a smooth curve.

3 Arc Length When a function y = f(x) is continuously differentiable on the interval [a, b], the arc length will be defined by the following way: For the partition a = x 0 < x 1 < x 2 < < x n = b the arc length can be approximated by using the sum of lengths of line segments: s = n i=1 n i=1 x i x i y i y i Δy i Δx i Using the limit we have n s = lim n i= Δy i Δx i (Δx i ). 2 (Δx i ) = a b 1 + f x 2 dx

4 Definition of Arc Length Let the function y = f(x) represent a smooth curve on the interval [a, b]. The arc length of f between a and b is s = a b 1 + f x 2 dx. Similarly, for a smooth curve x = g(y), the arc length of g between c and d is s = c d 1 + g y 2 dy.

5 Example 1. Find the arc length from (x 1, y 1 ) to (x 2, y 2 ) on the graph of f x = mx + b.

6 Example 2. Find the arc length of the graph y = x x on the interval 1 2, 2.

7 Example 3. Find the arc length of the graph y 1 3 = x 2 on the interval [0, 8].

8 Example 4. Find the arc length of the graph of y = ln(cos x) from x = 0 to x = π/4.

9 Example 5. An electric cable is hung between two towers that are 200 feet apart. The cable takes the shape of a catenary whose equation is y = 150 cosh x 150 = 75 ex/150 + e x/150 Find the arc length of the cable between the two towers.

10 Surface Area of Revolution When the graph of a continuous function is revolved about a line, the resulting surface is a surface of revolution. In computing the area of a surface of revolution the lateral surface area of the frustum of a right circular cone which has the following formula S = π r 1 + r 2 L = 2π 1 2 r 1 + r 2 L

11 Surface Area of Revolution Now consider a function f that has a continuous derivative on the interval [a, b]. Let Δ be a partition of [a, b], with subintervals of width Δx i. Then the line segment of length ΔL i = Δx i 2 + Δy i 2 generates a frustum of a cone.

12 Surface Area of Revolution Let r i be the average radius of this frustum. By Intermediate Value Theorem, a point d i exists in the ith subinterval such that r i = f(d i ). So the change of the surface area is ΔS i = 2πr i ܮΔ i = 2π f d i Δx i 2 + Δy i 2 = 2πf d i 1 + Δy i Δx i Now the surface area can be approximated by 2 Δx i n i=1 ΔS i = n i=1 2πf d i 1 + Δy i Δx i 2 Δx i.

13 By taking the limit n we have S = 2π a b f x 1 + f x 2 dx

14 Definition of the Area of a Surface of Revolution Let y = f(x) have a continuous derivative on the interval [a, b]. The area S of the surface of revolution formed by revolving the graph of f about a horizontal or vertical axis is S = 2π a b r x 1 + f x 2 dx where r(x) is the distance between the graph of f and the axis of revolution. If x = g(y) on the interval [c, d], then the surface area is S = 2π c d r y 1 + g y 2 dy where r(y) is the distance between the graph of g and the axis of revolution.

15 Example 6. Find the area of the surface formed by revolving the graph of f(x) = x 3 on the interval [0, 1] about the x-axis.

16 Example 7. Find the area of the surface formed by revolving the graph of f(x) = x 2 on the interval [0, 2] about the y-axis.

17 7.4 Work Work Done by a Constant Force If an object is moved a distance D in the direction of an applied constant force F, then the work W done by the force is defined as W = FD. There are four fundamental types of forces gravitational, electromagnetic, strong nuclear, and weak nuclear. A force can be thought of as a push or a pull; a force changes the state of rest or state of motion of a body. For gravitational forces on Earth, it is common to use units of measure corresponding to the weight of an object.

18 7.4 Work Example 1. Determine the work done in lifting a 50-pound object 4 feet. System of Measurement Measure of Work Measure of Force Measure of Distance U.S. foot-pound pound (l) foot (ft) International joule (J) newton (N) meter (m) C-G-S erg Dyne (dyn) centimeter (cm Conversions: 1 ft-lb J = ergs 1 N = 10 5 dyn lb 1 J = 10 7 ergs ft-lb 1 lb N

19 7.4 Work Work Done by a Variable Force If an object is moved along a straight line by a continuously varying force F(x), then the work W done by the force as the object is moved from x = a to x = b is given by W = a b F(x) dx. Idea: Let Δ be a partition of the interval: a = x 0 < x 1 < x 2 < < x n 1 < x n = b. We assume the force is constant as F c i (x i 1 c i x i ) for each interval. Then ΔW i = F c i Δx i So, the total work is approximated by Thus, W n i=1 n W = lim n i=1 W i = n i=1 F c i Δx i = F c i Δx i a b F(x) dx.

20 7.4 Work Some Laws in Physics 1. Hooke s Law: The force F required to compress or stretch a spring (within its elastic limits) is proportional to the distance d that the spring is compressed or stretched from its original length. That is, F = kd. Where the constant of proportionality k (the spring constant) depends on the specific nature of the spring.

21 7.4 Work Some Laws in Physics 2. Newton s Law of Universal Gravitation: The force F of attraction between two particles of masses m 1 and m 2 is proportional to the product of the masses and inversely proportional to the square of the distance d between the two particles. That is, F = G m 1m 2 d 2. When m 1 and m 2 are in kilograms and d in meters, F will be in newtons for a value of G = cubic meter per kilogram-second squared, where G is the gravitational constant.

22 7.4 Work Some Laws in Physics 3. Coulomb s Law: The force F between two charges q 1 and q 2 in a vacuum is proportional to the product of the charges and inversely proportional to the square of the distance d between the two charges. That is, F = k q 1q 2 d 2. When q 1 and q 2 are given in electrostatic units and d in centimeters, F will be in dynes for a value of k = 1.

23 7.4 Work Some Laws in Physics 1. Hooke s Law: The force F required to compress or stretch a spring (within its elastic limits) is proportional to the distance d that the spring is compressed or stretched from its original length. That is, F = kd. Where the constant of proportionality k (the spring constant) depends on the specific nature of the spring. Example 2. A force of 30 newtons compresses a spring 0.3 meter from its natural length of 1.5 meters. Find the work done in compressing the spring and additional 0.3 meter.

24 7.4 Work Example 4. A spherical tank of radius 8 feet is half full of oil that weights 50 pounds per cubic foot. Find the work required to pump all of the oil out through a hole in the top of the tank.

25 7.4 Work Example 5. A 20-foot chain weighting 5 pounds per foot is lying coiled on the ground. How much work is required to raise one end of the chain to a height of 20 feet so that it is fully extended.

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