Graphical Reasoning Consider the region bounded by the graphs of and where (a) Sketch a

In Exercises 1 4, find the center of mass of the point masses lying on the -axis. m1 6, m2 3, m3 5 x x1 5, x2 1, x3 3 m1

In Exercises 1 4, find the center of mass of the point masses lying on the -axis. m1 7, m2 4, m3 3, m4 8 x1 x1 3, x2 2, x3 5, x4 6 m1

In Exercises 1 4, find the center of mass of the point masses lying on the -axis. m1 1, m2 1, m3 1, m4 1, m5 1 x1 x1 7, x2 8, x3 12, x4 15, x5 18 m1

In Exercises 1 4, find the center of mass of the point masses lying on the -axis. m1 12, m2 1, m3 6, m4 3, m5 11 x1 x1 6, x2 4, x3 2, x4 0, x5 8 m1

Graphical Reasoning (a) Translate each point mass in Exercise 3 to the right five units and determine the resulting center of mass. (b) Translate each point mass in Exercise 4 to the left three units and determine the resulting center of mass.

Conjecture Use the result of Exercise 5 to make a conjecture about the change in the center of mass that results when each point mass is translated units horizontally. Statics Problems

In Exercises 7 and 8, consider a beam of length with a fulcrum feet from one end (see figure). There are objects with weights and placed on opposite ends of the beam. Find such that the system is in equilibrium. Two children weighing 50 pounds and 75 pounds are going to play on a seesaw that is 10 feet long.

In Exercises 7 and 8, consider a beam of length with a fulcrum feet from one end (see figure). There are objects with weights and placed on opposite ends of the beam. Find such that the system is in equilibrium. In order to move a 550-pound rock, a person weighing 200 pounds wants to balance it on a beam that is 5 feet long.

In Exercise 912, find the center of mass of the given system of point masses.

In Exercise 912, find the center of mass of the given system of point masses.

In Exercise 912, find the center of mass of the given system of point masses.

In Exercise 912, find the center of mass of the given system of point masses.

In Exercises 1324, find and for the laminas of uniform density bounded by the graphs of the equations. y x, y 0, x 4

In Exercises 1324, find and for the laminas of uniform density bounded by the graphs of the equations. y 1 2x2, y 0, x 2 y

In Exercises 1324, find and for the laminas of uniform density bounded by the graphs of the equations. y x2 , y x3 y

In Exercises 1324, find and for the laminas of uniform density bounded by the graphs of the equations. y x, y x y

In Exercises 1324, find and for the laminas of uniform density bounded by the graphs of the equations. y x2 4x 2, y x 2 y

In Exercises 1324, find and for the laminas of uniform density bounded by the graphs of the equations. y x 1, y 1 3x 1 y

In Exercises 1324, find and for the laminas of uniform density bounded by the graphs of the equations. y x23 , y 0, x 8 y

In Exercises 1324, find and for the laminas of uniform density bounded by the graphs of the equations. y x23 , y 4 y

In Exercises 1324, find and for the laminas of uniform density bounded by the graphs of the equations. x 4  y 2, x 0 y

In Exercises 1324, find and for the laminas of uniform density bounded by the graphs of the equations. x 2y  y 2, x 0 x

In Exercises 1324, find and for the laminas of uniform density bounded by the graphs of the equations. x y, x 2y  y 2 x

In Exercises 1324, find and for the laminas of uniform density bounded by the graphs of the equations. x y 2, x y 2 x

In Exercises 25 28, set up and evaluate the integrals for finding the area and moments about the - and -axes for the region bounded by the graphs of the equations. (Assume ) y x2 , y x

In Exercises 25 28, set up and evaluate the integrals for finding the area and moments about the - and -axes for the region bounded by the graphs of the equations. (Assume ) y 1 x , y 0, 1 x 4 y

In Exercises 25 28, set up and evaluate the integrals for finding the area and moments about the - and -axes for the region bounded by the graphs of the equations. (Assume ) y 2x 4, y 0, 0 x 3 y

In Exercises 25 28, set up and evaluate the integrals for finding the area and moments about the - and -axes for the region bounded by the graphs of the equations. (Assume ) y x2  4, y 0 y

In Exercises 29 32, use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. y 10x125  x3 , y 0 y

In Exercises 29 32, use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. y xex2 , y 0, x 0, x 4 y 1

In Exercises 29 32, use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. Prefabricated End Section of a Building Prefabricated End Section of a Building y 5 3 400 x2 , y 0 y

In Exercises 29 32, use a graphing utility to graph the region bounded by the graphs of the equations. Use the integration capabilities of the graphing utility to approximate the centroid of the region. Witch of Agnesi Witch of Agnesi y 8x2 4 , y 0, x 2, x 2 y 5

In Exercises 3338, find and/or verify the centroid of the common region used in engineering. Triangle Show that the centroid of the triangle with vertices and is the point of intersection of the medians (see figure). Figure for 33 Figure for 34

In Exercises 3338, find and/or verify the centroid of the common region used in engineering. Parallelogram Show that the centroid of the parallelogram with vertices and is the point of intersection of the diagonals (see figure).

In Exercises 3338, find and/or verify the centroid of the common region used in engineering. Trapezoid Find the centroid of the trapezoid with vertices and Show that it is the intersection of the line connecting the midpoints of the parallel sides and the line connecting the extended parallel sides, as shown in the figure. Figure for 35 Figure for 36

In Exercises 3338, find and/or verify the centroid of the common region used in engineering. Semicircle Find the centroid of the region bounded by the graphs of and (see figure).

In Exercises 3338, find and/or verify the centroid of the common region used in engineering. Semiellipse Find the centroid of the region bounded by the graphs of and (see figure). Figure for 37 Figure for 38

In Exercises 3338, find and/or verify the centroid of the common region used in engineering. Parabolic Spandrel Find the centroid of the parabolic spandrel shown in the figure.

Graphical Reasoning Consider the region bounded by the graphs of and where (a) Sketch a graph of the region. (b) Use the graph in part (a) to determine Explain. (c) Set up the integral for finding Because of the form of the integrand, the value of the integral can be obtained without integrating. What is the form of the integrand and what is the value of the integral? Compare with the result in part (b). (d) Use the graph in part (a) to determine whether or Explain. (e) Use integration to verify your answer in part (d).

Graphical and Numerical Reasoning Consider the region bounded by the graphs of and where and is a positive integer. (a) Set up the integral for finding Because of the form of the integrand, the value of the integral can be obtained without integrating. What is the form of the integrand and what is the value of the integral? Compare with the result in part (b). (b) Is or Explain. (c) Use integration to find as a function of (d) Use the result of part (c) to complete the table. (e) Find (f ) Give a geometric explanation of the result in part (e).

Modeling Data The manufacturer of glass for a window in a conversion van needs to approximate its center of mass. A coor- dinate system is superimposed on a prototype of the glass (see figure). The measurements (in centimeters) for the right half of the symmetric piece of glass are shown in the table. (a) Use Simpsons Rule to approximate the center of mass of the glass. (b) Use the regression capabilities of a graphing utility to find a fourth-degree polynomial model for the data. (c) Use the integration capabilities of a graphing utility and the model to approximate the center of mass of the glass. Compare with the result in part (a).

Modeling Data The manufacturer of a boat needs to approx- imate the center of mass of a section of the hull. A coordinate system is superimposed on a prototype (see figure). The measurements (in feet) for the right half of the symmetric prototype are listed in the table. (a) Use Simpsons Rule to approximate the center of mass of the hull section. (b) Use the regression capabilities of a graphing utility to find fourth-degree polynomial models for both curves shown in the figure. Plot the data and graph the models. (c) Use the integration capabilities of a graphing utility and the model to approximate the center of mass of the hull section. Compare with the result in part (a).

In Exercises 43 46, introduce an appropriate coordinate system and find the coordinates of the center of mass of the planar lamina. (The answer depends on the position of the coordinate system.)

In Exercises 43 46, introduce an appropriate coordinate system and find the coordinates of the center of mass of the planar lamina. (The answer depends on the position of the coordinate system.)

In Exercises 43 46, introduce an appropriate coordinate system and find the coordinates of the center of mass of the planar lamina. (The answer depends on the position of the coordinate system.)

In Exercises 43 46, introduce an appropriate coordinate system and find the coordinates of the center of mass of the planar lamina. (The answer depends on the position of the coordinate system.)

Find the center of mass of the lamina in Exercise 43 if the circular portion of the lamina has twice the density of the square portion of the lamina.

Find the center of mass of the lamina in Exercise 43 if the square portion of the lamina has twice the density of the circular portion of the lamina.

In Exercises 4952, use the Theorem of Pappus to find the volume of the solid of revolution. The torus formed by revolving the circle about the axis

In Exercises 4952, use the Theorem of Pappus to find the volume of the solid of revolution. The torus formed by revolving the circle about the axis

In Exercises 4952, use the Theorem of Pappus to find the volume of the solid of revolution. The solid formed by revolving the region bounded by the graphs of and about the axis

In Exercises 4952, use the Theorem of Pappus to find the volume of the solid of revolution. The solid formed by revolving the region bounded by the graphs of and about the axis

Let the point masses be located at Define the center of mass

What is a planar lamina? Describe what is meant by the center of mass of a planar lamina.

The centroid of the plane region bounded by the graphs of and is Is it possible to find the centroid of each of the regions bounded by the graphs of the following sets of equations? If so, identify the centroid and explain your answer. (a) and (b) and (c) and (d) and

State the Theorem of Pappus.

In Exercises 57 and 58, use the Second Theorem of Pappus , which is stated as follows. If a segment of a plane curve is revolved about an axis that does not intersect the curve (except possibly at its endpoints), the area of the resulting surface of revolution is given by the product of the length of times the distance traveled by the centroid of . A sphere is formed by revolving the graph of about the axis. Use the formula for surface area, to find the centroid of the semicircle

In Exercises 57 and 58, use the Second Theorem of Pappus , which is stated as follows. If a segment of a plane curve is revolved about an axis that does not intersect the curve (except possibly at its endpoints), the area of the resulting surface of revolution is given by the product of the length of times the distance traveled by the centroid of . A torus is formed by revolving the graph of about the axis. Find the surface area of the torus.

Let be constant, and consider the region bounded by the axis, and Find the centroid of this region. As what does the region look like, and where is its centroid?

Let be the region in the cartesian plane consisting of all points satisfying the simultaneous conditions and Find the centroid of