Calculus
7th Edition
ISBN: 9781524916817
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Question
Chapter 12.6, Problem 10PS
To determine
To find: the centroid for the regions of the solid
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Problem 2: Write down integrals (without evaluating them) representing the volumes of
the solids resulting from rotating the region enclosed by x = 4y – y? and x = 0 around:
a. the y-axis.
b. the linex = -1.
Problems:
1. Find the centroid for the area surrounded by the curve y=x', y-axis
and the line y=8 ?
2. Find the centroid of the area enclosed between the line
y=10-2x
and the two Cartesian axis?
3. The area surrounded by the curve y=2x²-x+6, y-axis and the line y=9
represents a part of a solid area .Find its centroid?
4. Find the centroid of the shape shown in figure?
7
4
4
4
PRACTICE PROBLEMS
1. For the shaded region shown, determine the
following using vertical strip:
Y
a. Area
x= 2y - 3
4
b. First moment of the shaded area with
respect to x-axis
c. First moment of the shaded area with
:-:
respect to y-axis
d. Centroid (x, y)
1
-1
-1 0 1 2 3 4 5
343/48 units?; Q, = 2401/240
units; Q, = 1029/64 units³; (x, y)= (9/4, 7/5)
Answer: Area
%3D
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. 59PSCh. 12.1 - Prob. 60PSCh. 12.2 - Prob. 1PSCh. 12.2 - Prob. 2PSCh. 12.2 - Prob. 3PSCh. 12.2 - Prob. 4PSCh. 12.2 - Prob. 5PSCh. 12.2 - Prob. 6PSCh. 12.2 - Prob. 7PSCh. 12.2 - Prob. 8PSCh. 12.2 - Prob. 9PSCh. 12.2 - Prob. 10PSCh. 12.2 - Prob. 11PSCh. 12.2 - Prob. 12PSCh. 12.2 - Prob. 13PSCh. 12.2 - Prob. 14PSCh. 12.2 - Prob. 15PSCh. 12.2 - Prob. 16PSCh. 12.2 - Prob. 17PSCh. 12.2 - Prob. 18PSCh. 12.2 - Prob. 19PSCh. 12.2 - Prob. 20PSCh. 12.2 - Prob. 21PSCh. 12.2 - Prob. 22PSCh. 12.2 - Prob. 23PSCh. 12.2 - Prob. 24PSCh. 12.2 - Prob. 25PSCh. 12.2 - Prob. 26PSCh. 12.2 - Prob. 27PSCh. 12.2 - Prob. 28PSCh. 12.2 - Prob. 29PSCh. 12.2 - Prob. 30PSCh. 12.2 - Prob. 31PSCh. 12.2 - Prob. 32PSCh. 12.2 - Prob. 33PSCh. 12.2 - Prob. 34PSCh. 12.2 - Prob. 35PSCh. 12.2 - Prob. 36PSCh. 12.2 - Prob. 37PSCh. 12.2 - Prob. 38PSCh. 12.2 - Prob. 39PSCh. 12.2 - Prob. 40PSCh. 12.2 - Prob. 41PSCh. 12.2 - Prob. 42PSCh. 12.2 - Prob. 43PSCh. 12.2 - Prob. 44PSCh. 12.2 - Prob. 45PSCh. 12.2 - Prob. 46PSCh. 12.2 - Prob. 47PSCh. 12.2 - 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Prob. 37PSCh. 12.3 - Prob. 38PSCh. 12.3 - Prob. 39PSCh. 12.3 - Prob. 40PSCh. 12.3 - Prob. 41PSCh. 12.3 - Prob. 42PSCh. 12.3 - Prob. 43PSCh. 12.3 - Prob. 44PSCh. 12.3 - Prob. 45PSCh. 12.3 - Prob. 46PSCh. 12.3 - Prob. 47PSCh. 12.3 - Prob. 48PSCh. 12.3 - Prob. 49PSCh. 12.3 - Prob. 50PSCh. 12.3 - Prob. 51PSCh. 12.3 - Prob. 52PSCh. 12.3 - Prob. 53PSCh. 12.3 - Prob. 54PSCh. 12.3 - Prob. 55PSCh. 12.3 - Prob. 56PSCh. 12.3 - Prob. 57PSCh. 12.3 - Prob. 58PSCh. 12.3 - Prob. 59PSCh. 12.3 - Prob. 60PSCh. 12.4 - Prob. 1PSCh. 12.4 - Prob. 2PSCh. 12.4 - Prob. 3PSCh. 12.4 - Prob. 4PSCh. 12.4 - Prob. 5PSCh. 12.4 - Prob. 6PSCh. 12.4 - Prob. 7PSCh. 12.4 - Prob. 8PSCh. 12.4 - Prob. 9PSCh. 12.4 - Prob. 10PSCh. 12.4 - Prob. 11PSCh. 12.4 - Prob. 12PSCh. 12.4 - Prob. 13PSCh. 12.4 - Prob. 14PSCh. 12.4 - Prob. 15PSCh. 12.4 - Prob. 16PSCh. 12.4 - Prob. 17PSCh. 12.4 - Prob. 18PSCh. 12.4 - Prob. 19PSCh. 12.4 - Prob. 20PSCh. 12.4 - Prob. 21PSCh. 12.4 - Prob. 22PSCh. 12.4 - Prob. 23PSCh. 12.4 - Prob. 24PSCh. 12.4 - Prob. 25PSCh. 12.4 - 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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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- Work Problem 2 a) Find the points of intersection of the curves y = -52² and y=z² - 6. b) Find the Volume of the solid obtained by rotating the region bounded by the curves about the z-axis. Work Problem 3 Find the exact length L of the curve where y 20. y=-52² and y=2²-6, 7y² = (3x-2)³, 1 ≤z≤ 2,arrow_forward2. Suppose that the triangle with vertices (-1,-1), (0, 1) and (1,–1) is revolved about the y – axis. Find the volume of the resulting solid.arrow_forwardVx² + y² and inside the sphere x² + y² + z² = 9. Problem 1: Let E be the solid above the cone z = Find the volume of E.arrow_forward
- 3. 1. Use the method of disksirings to determine the volume of the solid obtained by rotating the region bounded by y= *, y= 2, z and a3 about the (a) line y 7 (b) line y =1 (c) line y= -3 Jearrow_forwardProblem 3. Use a double integral to find the volume in the first octant bounded by the coordinate planes, the plane y = 4, and the plane z = 5 3*.arrow_forward55. The region in the first quadrant, which is bounded by the curve y² = 4x, and the line x = 4, is revolved about the line x = 4, Locate the centroid of the resulting solid of revolution. A. 1.25 units B. 2 units C. 1.50 units D. 1 unitarrow_forward
- 8. Find the centroid of the first quadrant area bounded by x2 + y - 4 = 0 9 Find the centroid of the first and secondarrow_forward53. The region in the first quadrant, which is bounded by the curve x² = 4y, the line x = 4, is revolved about the line x = 4. Locate the centroid of the resulting solid of revolution. A. 0.8 B. 0.5 C. 1 D. 0.6arrow_forward4. Let E be the region in the first quadrant bounded by the graphs of f(x) = Vx – 1, y = 2, and x = 1, as shown in the figure above. (a) There is a value of c for which the vertical line x = c divides E into two regions with equal area. The value of c is (b) The volume of the solid generated by rotating E around the x-axis is (C) The volume of the solid generated by rotating E around the y-axis is (d) Region E is the base of a solid. Cross sections of the solid perpendicular to the x-axis are squares. The volume of this solid isarrow_forward
- 3. Consider the region R bounded by the curves y = In(x)+1, y = -x+2, and the vertical line with equation x = 5. (The three curves are given below.) Find the volume of the solid R obtained by rotating the region R about the vertical line with equation x = -1. (You might use that In(x)+1= -x+2 if and only if x = = 1.) g(æ) = -x+2 x = 5 -2 -1 8 10 11 12 -2 f(x) = In(x}+1arrow_forwardFind the volume V generated by rotated the region A about the x-axis as shown in the Figure 1. 30) Figure 1 for Problem 30 Figure 2 for Problem 31 Figure 3 for Problem 32 ze 6 fix)= 6-x² 4 2. * go)=2 0,5 A = ? A= ? V=? Region A is bounded by ftop(x) = 2e2x, and gbot(x) =1, over [0, 0. 5]. V=, TI-Calculator, Window Xmin=0, Xmax =0. 5, Ymin = 0, Ymax = 30] A) 3. 89t [Hint By B) 4. 89T C) 5. 89n D) 6. 89T Solve the problem. 31) Find the shaded Area A between two curves ftop(x) = 6 – x2 and gbot(x) = 2, over [0, 2] as shown in the Figure 2 above. A) A =16/3 B) A =15/2 C) A =17/3 D) A =27/5 32) Find the shaded Area A under the curve f(x) = 3× ln 3 over [1, 3] as shown in the Figure 3 above. A) A = 24 B) A = 27 C) A = 29ln3 D) A = 27ln3arrow_forward6. Find the volume of the solid under the surface (of the paraboloid) z = xy and above the triangle whose vertices are (0, 1), (1, 1) and (1,2).arrow_forward
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