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 <=t<=b, 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 x=f(t) and y=g(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. Pappus’s 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(X=x), 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(a<=X<=b) = F(b) – F(a)