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 16PS
To determine
To find: the center of mass of a lamina by using the double
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Consider 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 .
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..
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?
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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Prob. 3PSCh. 12.6 - Prob. 4PSCh. 12.6 - Prob. 5PSCh. 12.6 - Prob. 6PSCh. 12.6 - Prob. 7PSCh. 12.6 - Prob. 8PSCh. 12.6 - Prob. 9PSCh. 12.6 - Prob. 10PSCh. 12.6 - Prob. 11PSCh. 12.6 - Prob. 12PSCh. 12.6 - Prob. 13PSCh. 12.6 - Prob. 14PSCh. 12.6 - Prob. 15PSCh. 12.6 - Prob. 16PSCh. 12.6 - Prob. 17PSCh. 12.6 - Prob. 18PSCh. 12.6 - Prob. 19PSCh. 12.6 - Prob. 20PSCh. 12.6 - Prob. 21PSCh. 12.6 - Prob. 22PSCh. 12.6 - Prob. 23PSCh. 12.6 - Prob. 24PSCh. 12.6 - Prob. 25PSCh. 12.6 - Prob. 26PSCh. 12.6 - Prob. 27PSCh. 12.6 - Prob. 28PSCh. 12.6 - Prob. 29PSCh. 12.6 - Prob. 30PSCh. 12.6 - Prob. 31PSCh. 12.6 - Prob. 32PSCh. 12.6 - Prob. 33PSCh. 12.6 - Prob. 34PSCh. 12.6 - Prob. 35PSCh. 12.6 - Prob. 36PSCh. 12.6 - Prob. 37PSCh. 12.6 - Prob. 38PSCh. 12.6 - Prob. 39PSCh. 12.6 - Prob. 40PSCh. 12.6 - Prob. 41PSCh. 12.6 - Prob. 42PSCh. 12.6 - Prob. 43PSCh. 12.6 - Prob. 44PSCh. 12.6 - Prob. 45PSCh. 12.6 - Prob. 46PSCh. 12.6 - Prob. 47PSCh. 12.6 - Prob. 48PSCh. 12.6 - Prob. 49PSCh. 12.6 - Prob. 50PSCh. 12.6 - 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- 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_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_forward.2п 2 5²π S²² S ² zr√r² + z²dzdrd0 •2πT 2 S²π S²² S ₂² r √√r² + z²dzdrdə 52²arrow_forward
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- 1.Find the area enclosed by the curve y=x⁴ and y²=8x 2.find the area enclosed by the curve y=x²+1 and y=7-xarrow_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_forward1. The region in the first quadrant bounded by y = x², y= 0, and x = 3 is rotated about the line y = -1. SET-UP the appropriate integral to find the volume of the resulting solid using: (a) dx. (b) dy. (c) Now, find the exact volume of the solid.arrow_forward
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