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Chapter 5

- Applications of the Integral

5.1 Area of a Plane Region

- Definite integral from a to b is the area

contained between f(x) and the x-axis on that

interval. - Area between two curves is found by 1)

determining where the 2 functions intersect, 2)

determining which function is the greater

function over that interval, and 3) evaluating

the definite integral over the interval of

greater function minus lesser function.

Find the area enclosed by the functions

5.2 Volumes of Solids Slabs, Disks, Washers

Solids of Revolution Disk Method

- A solid may be formed by revolving a curve about

an axis. - The volume of this solid may be found by

considering the solid sliced into many, many

round disks. - The area of each disk is the area of a circle.

Volume is found by integrating the area. The

radius of each circle is f(x) for each x value in

the interval.

Washer method

- If the area between two curves is revolved around

an axis, a solid is created that is hollow in

the center. - When slicing this solid, the sections created are

washers, not solid circles. - The area of the smaller circle must be subtracted

from the area of the larger one.

5.3 Volumes of Solids of Revolutions Shells

- When an area between two curves is revolved about

an axis a solid is created. - This solid could be considered as the sum of

many, many concentric cylinders. - Volume is the integral of the area, in this case

it is the surface area of the cylinder, thus r

x and h f(x)

Does it matter which method to use?

- Either method may work. Sketch a picture of the

function to determine which method may be easier. - If a specific method is requested, that method

should be implemented.

5.4 Length of a Plane Curve

- A plane curve is smooth if it is determined by a

pair of parametric equations x f(t) and y

g(t), a lttltb, where f and g exist and are

continuous on a,b, and f(t) and g(t) are not

simultaneously zero on (a,b). - If the curve is smooth, we can find its length.

Approximate curve length by the sum of many, many

line segments.

- To have the actual length, you would need

infinitely many line segments, each whose length

is found using the Pythagorean theorem. - The length of a smooth curve, defined as xf(t)

and yg(t) is

What if the function is not parametric, but

defined as y f(x)?

- Infinitely many line segments still provide the

length. Again, use the Pythagorean formula with

horizontal component x and vertical component

dy/dx for every line segment.

5.5 Work Fluid Force

- Work Force x Distance
- In many cases, the force is not constant

throughout the entire distance. - To determine total work done, add all the amounts

of work done throughout the interval INTEGRATE! - If the force is defined as F(x), then work is

Fluid Force

- If a tank is filled to a depth h with a fluid of

density (sigma), then the force exerted by the

fluid on a horizontal rectangle of area A on the

bottom is equal to the weight of the column of

fluid that stands directly over that rectangle. - Let sigma density, h(x)depth, w(x)width, then

force is

5.6 Moments and Center of Mass

- The product of the mass m of a particle and its

directed distance from a point (its lever arm) is

called the moment of the particle with respect to

that point. It measures the tendency of the mass

to produce a rotation about the point. - 2 masses along a line balance at a point if the

sum of their moments with respect to that point

is zero. - The center of mass is the balance point.

Finding the center of mass let M moment, m

mass, sigma density

Centroid For a planar region, the center of

pass of a homogeneous lamina is the centroid.

- Pappuss Theorem If a region R, lying on one

side of a line in its plane, is revolved about

that line, then the volume of the resulting solid

is equal to the area of R multiplied by the

distance traveled by its centroid.

5.7 Probability and Random Variables

- Expectation of a random variable If X is a

random variable with a given probability

distribution, p(Xx), then the expectation of X,

denoted E(X), also called the mean of X and

denoted as mu, is

Probability Density Function (PDF)

- If the outcomes are not finite (discrete), but

could be any real number in an interval, it is

continuous. - Continuous random variables are studied similarly

to distribution of mass. - The expected value (mean) of a continuous random

variable X is

Theorem A

- Let X be a continuous random variable taking on

values in the interval A,B and having PDF f(x)

and CDF (cumulative distribution function) F(x).

Then - 1. F(x) f(x)
- 2. F(A) 0 and F(B) 1
- 3. P(altXltb) F(b) F(a)

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