Calculus
6th Edition
ISBN: 9781465208880
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 12.6, Problem 17PS
To determine
To find: the center of mass of a lamina by using the double
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3. Find the mass and center of the following lamina: The region bounded by y = sin x and y = 1 – sin x
between x = 1 and x
= * with constant density p(x, y) = 1..
The triangular region Ω has vertices (1, 1), (2, 0), and (2, 3).
1. Let S be the region bounded by the graphs of y = In x, y = 0, x = 2.
(a) Sketch carefully this region.
(b) Calculate the volume of the body obtained by rotating S around the y-axis by using the
shell method.
You will need to use integration by parts. State what u, dv, du, v are. Do not skip any
details.
(c) Calculate the volume of the body obtained by rotating S around the x-axis by using the
disk method.
You will need to use integration by parts. State what u, dv, du, v are. Do not skip any
details.
Chapter 12 Solutions
Calculus
Ch. 12.1 - Prob. 1PSCh. 12.1 - Prob. 2PSCh. 12.1 - Prob. 3PSCh. 12.1 - Prob. 4PSCh. 12.1 - Prob. 5PSCh. 12.1 - Prob. 6PSCh. 12.1 - Prob. 7PSCh. 12.1 - Prob. 8PSCh. 12.1 - Prob. 9PSCh. 12.1 - Prob. 10PS
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- 2. Find the exact location of the center of mass (x,y), given the region bounded by the parabola y = x² - 4x + 4 and the line y = 4. 3. Suppose an ant travels from point A(-1 cm, 7 cm) to point B(-8 cm, 25 cm). How far would it travel along the curve y = 6x²/3 + 1?arrow_forwardConsider the plate bounded by y² = 8x and x = 2 with density d = 2 - x. Note: When dealing with a curve like y² = 8x, it's important to note that there can be negative y-values. In particular, if taking square roots (which is not necessary), be sure to include both positive and negative square roots. 1. Sketch a graph of the plate. Shade in the region and number the axes so that the important points on the plate can be read from the graph. 2. Express the mass of the plate as a double integral and evaluate to find the mass. 3. Express the x-coordinate¹ of the center of mass of the plate as a double integral and evaluate to find the value of .arrow_forward-2. Let S be the solid obtained by rotating the region bounded by the curves y = x + (9/x), y= 10 , about x = dx , where a = and 6 = The volume V = Therefore the volume V =arrow_forward
- 3. Suppose you have a solid, flat plate in the shape of a region R with density function 6(x, y) kg/m². You are given the following information: R J = 8 (z, y) dA = 10 16 3 [[₂y 8(x, y) dA = 20 SS 6(x, y) dA = 4 R R We can conclude that the mass of the plate is kg, and the center of mass has coordinates (,7) = (arrow_forward4. Perform the following (a) A gold-plated plate D is defined by the rectangle enclosed by the lines x = 0, x = 2π, y = 0, and y = π (in centimeters) and has a mass density of 8(x, y) = y² sin²(4x) + 2 grams per square centimeter. If the gold costs 7 USD per gram, how much is the gold present on the plate worth? (b) Find the average value of the function f(x, y, z) = e¯² over the sphere x² + y² + z² ≤ 1.arrow_forward2. Find the mass of the lamina bounded by the graphs of y= the density function is p(x, y) = kx^y. 1 x2¹ y = 0, x= 1, and z = 4 ifarrow_forward
- 2. The closed region between the curves y² = 72 and y2 = 10 x in the first quadrant is rotated about the z-axis. 4 (a) The x-coordinate of the point where the curves intersect in the first quadrant is (b) The volume of the resultant solid of revolution is 5arrow_forwardThe velocity field in the region shown is given by V = vỹ = ( aj + byk ) where a = 10 m/s and b = 5 s-1. For the 1 m × 1 m triangular control volume (depth w = 1 m perpendicular to the diagram), an element of area (1 may be represented by dA1 = wdzj– wdyk and an element of area (2 by dA2 = -wdyk. Z. 2 Control volume → y P4.4 a. Find an expression for V · dA1. b. Evaluate Sa, V. dA1. c. Find an expression for V · dA2. d. Find an expression for V (v. dAz). e. Evaluate · dA2arrow_forward(b) insects per cubic foot. Describe the set of all points at which the insect density is 20 (c) shape of the level surfaces? Describe the level surfaces of the function p. Identify the generalarrow_forward
- how do i do problem 5?arrow_forward1. Let the joint pdf of X and Y be given by fx.y (r, y) Cexp 1 + ry y) 1(0,00) (x + y) where C > 0 is an unknown constant. Hint: this question encourages a geometric approach. In your integrations, you may wish to carry out a change of coordinates so that the new coordinate axes, while still being perpendicular, align well with the area sketched in part (a) below. (a) Sketch the area in the plane R? where fx,y is positive. (b) Do X, Y jointly follow a normal distribution? Justify your answer. (c) Decide whether g(x, y) = C' exp (-a2 + ry - u) defines a pdf on R2 (Note that g is only one factor of fxy, not the whole of fx.y). If yes, compute C". If no, show that there is no C" > 0 for which it is a pdf. Hint: Consider the change of coordinates w = x +y, z =x - y. (d) Compute the constant C.arrow_forward1. A particular tank (on the surface of the Earth) is an inverted right circular cone, sticking 10 meters down into the ground with a circular top of radius 4 meters. The bottom of the tank has 5 meters of fluid. The density of the fluid is given by 8(h) =h² kg/m³ at a distance h meters down from the top of the tank (so h=0 is the top of the tank and h= 10 is the bottom of the tank). The work required to pump the fluid out the top of the tank can be written as an integral of the form ff(h)dh. (a) What is the lower limit of integration, a? (b) What is the upper limit of integration, b? (c) What is the integrand, f(h)? (d) What are the units of work for the integral? (Pick one from the list below.).arrow_forward
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