Calculus
6th Edition
ISBN: 9781465208880
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 6.2, Problem 24PS
To determine
To find: The volume of the solid formed by revolving the region inside the ellipse.
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Students have asked these similar questions
1. Find the volume of an ellipsoid obtained by rotating the region confined between the y-axis and
the graph of ellipse
x2
= 1.
62
a2
(a) Sketch carefully the graph of an ellipse. Feel free to google it, but you should learn how to
provide a plot of an ellipse given arbitrary positive constants a and b.
(b) Identify the region that you will rotate around the y-axis and show it on your sketch.
(c) Use the method of disks to find the volume (set up the integral and evaluate it providing
all the details)..
(d) Use the method of cylindrical shells to find the volume (set up the integral and evaluate it
providing all the details).
(e) Did you get the same answers in the two previous points?
(f) Make a correct guess without any calculations about the volume of an ellipsoid obtained by
rotating around the x-axis the region between the x-axis and the graph of the same ellipse.
1. Let a, b > 0.
(a) Calculate the area inside the ellipse given by the equation
x2
y²
1.
a2
62
(b) Calculate the volume of the solid obtained by revolving the upper half of the
ellipse from part a) about the x-axis.
4. A solid has a circular base with equation x² + y² = 9. If every cross section perpendicular to the y-axis is an
equilateral triangle, then its volume is
(A) 36
(B) 12/3
(C) 18/3
(D) 24/3
(E) 36/3
Chapter 6 Solutions
Calculus
Ch. 6.1 - Prob. 1PSCh. 6.1 - Prob. 2PSCh. 6.1 - Prob. 3PSCh. 6.1 - Prob. 4PSCh. 6.1 - Prob. 5PSCh. 6.1 - Prob. 6PSCh. 6.1 - Prob. 7PSCh. 6.1 - Prob. 8PSCh. 6.1 - Prob. 9PSCh. 6.1 - Prob. 10PS
Ch. 6.1 - Prob. 11PSCh. 6.1 - Prob. 12PSCh. 6.1 - Prob. 13PSCh. 6.1 - Prob. 14PSCh. 6.1 - Prob. 15PSCh. 6.1 - Prob. 16PSCh. 6.1 - Prob. 17PSCh. 6.1 - Prob. 18PSCh. 6.1 - Prob. 19PSCh. 6.1 - Prob. 20PSCh. 6.1 - Prob. 21PSCh. 6.1 - Prob. 22PSCh. 6.1 - Prob. 23PSCh. 6.1 - Prob. 24PSCh. 6.1 - Prob. 25PSCh. 6.1 - Prob. 26PSCh. 6.1 - Prob. 27PSCh. 6.1 - Prob. 28PSCh. 6.1 - Prob. 29PSCh. 6.1 - Prob. 30PSCh. 6.1 - Prob. 31PSCh. 6.1 - Prob. 32PSCh. 6.1 - Prob. 33PSCh. 6.1 - Prob. 34PSCh. 6.1 - Prob. 35PSCh. 6.1 - Prob. 36PSCh. 6.1 - Prob. 37PSCh. 6.1 - Prob. 38PSCh. 6.1 - Prob. 39PSCh. 6.1 - Prob. 40PSCh. 6.1 - Prob. 41PSCh. 6.1 - Prob. 42PSCh. 6.1 - Prob. 43PSCh. 6.1 - Prob. 44PSCh. 6.1 - Prob. 45PSCh. 6.1 - Prob. 46PSCh. 6.1 - Prob. 47PSCh. 6.1 - Prob. 48PSCh. 6.1 - Prob. 49PSCh. 6.1 - Prob. 50PSCh. 6.1 - Prob. 51PSCh. 6.1 - Prob. 52PSCh. 6.1 - Prob. 53PSCh. 6.1 - Prob. 54PSCh. 6.1 - Prob. 55PSCh. 6.1 - Prob. 56PSCh. 6.1 - Prob. 57PSCh. 6.1 - Prob. 58PSCh. 6.1 - Prob. 59PSCh. 6.1 - Prob. 60PSCh. 6.2 - Prob. 1PSCh. 6.2 - Prob. 2PSCh. 6.2 - Prob. 3PSCh. 6.2 - Prob. 4PSCh. 6.2 - Prob. 5PSCh. 6.2 - Prob. 6PSCh. 6.2 - Prob. 7PSCh. 6.2 - Prob. 8PSCh. 6.2 - Prob. 9PSCh. 6.2 - Prob. 10PSCh. 6.2 - Prob. 11PSCh. 6.2 - Prob. 12PSCh. 6.2 - Prob. 13PSCh. 6.2 - Prob. 14PSCh. 6.2 - Prob. 15PSCh. 6.2 - Prob. 16PSCh. 6.2 - Prob. 17PSCh. 6.2 - Prob. 18PSCh. 6.2 - Prob. 19PSCh. 6.2 - Prob. 20PSCh. 6.2 - Prob. 21PSCh. 6.2 - Prob. 22PSCh. 6.2 - Prob. 23PSCh. 6.2 - Prob. 24PSCh. 6.2 - Prob. 25PSCh. 6.2 - Prob. 26PSCh. 6.2 - Prob. 27PSCh. 6.2 - Prob. 28PSCh. 6.2 - Prob. 29PSCh. 6.2 - Prob. 30PSCh. 6.2 - Prob. 31PSCh. 6.2 - Prob. 32PSCh. 6.2 - Prob. 33PSCh. 6.2 - Prob. 34PSCh. 6.2 - Prob. 35PSCh. 6.2 - Prob. 36PSCh. 6.2 - Prob. 37PSCh. 6.2 - Prob. 38PSCh. 6.2 - Prob. 39PSCh. 6.2 - Prob. 40PSCh. 6.2 - Prob. 41PSCh. 6.2 - Prob. 42PSCh. 6.2 - Prob. 43PSCh. 6.2 - Prob. 44PSCh. 6.2 - Prob. 45PSCh. 6.2 - Prob. 46PSCh. 6.2 - Prob. 47PSCh. 6.2 - Prob. 48PSCh. 6.2 - Prob. 49PSCh. 6.2 - Prob. 50PSCh. 6.2 - Prob. 51PSCh. 6.2 - Prob. 52PSCh. 6.2 - Prob. 53PSCh. 6.2 - Prob. 54PSCh. 6.2 - Prob. 55PSCh. 6.2 - Prob. 56PSCh. 6.2 - Prob. 57PSCh. 6.2 - Prob. 58PSCh. 6.2 - Prob. 59PSCh. 6.2 - Prob. 60PSCh. 6.3 - Prob. 1PSCh. 6.3 - Prob. 2PSCh. 6.3 - Prob. 3PSCh. 6.3 - Prob. 4PSCh. 6.3 - Prob. 5PSCh. 6.3 - Prob. 6PSCh. 6.3 - Prob. 7PSCh. 6.3 - Prob. 8PSCh. 6.3 - Prob. 9PSCh. 6.3 - Prob. 10PSCh. 6.3 - Prob. 11PSCh. 6.3 - Prob. 12PSCh. 6.3 - Prob. 13PSCh. 6.3 - Prob. 14PSCh. 6.3 - Prob. 15PSCh. 6.3 - Prob. 16PSCh. 6.3 - Prob. 17PSCh. 6.3 - Prob. 18PSCh. 6.3 - Prob. 19PSCh. 6.3 - Prob. 20PSCh. 6.3 - Prob. 21PSCh. 6.3 - Prob. 22PSCh. 6.3 - Prob. 23PSCh. 6.3 - Prob. 24PSCh. 6.3 - Prob. 25PSCh. 6.3 - Prob. 26PSCh. 6.3 - Prob. 27PSCh. 6.3 - Prob. 28PSCh. 6.3 - Prob. 29PSCh. 6.3 - Prob. 30PSCh. 6.3 - Prob. 31PSCh. 6.3 - Prob. 32PSCh. 6.3 - Prob. 33PSCh. 6.3 - Prob. 34PSCh. 6.3 - Prob. 35PSCh. 6.3 - Prob. 36PSCh. 6.3 - Prob. 37PSCh. 6.3 - Prob. 38PSCh. 6.3 - Prob. 39PSCh. 6.3 - Prob. 40PSCh. 6.3 - Prob. 41PSCh. 6.3 - Prob. 42PSCh. 6.3 - Prob. 43PSCh. 6.3 - Prob. 44PSCh. 6.3 - Prob. 45PSCh. 6.3 - Prob. 46PSCh. 6.3 - Prob. 47PSCh. 6.3 - Prob. 48PSCh. 6.3 - Prob. 49PSCh. 6.3 - Prob. 50PSCh. 6.3 - Prob. 51PSCh. 6.3 - Prob. 52PSCh. 6.3 - Prob. 53PSCh. 6.3 - Prob. 54PSCh. 6.3 - Prob. 55PSCh. 6.3 - Prob. 56PSCh. 6.3 - Prob. 57PSCh. 6.3 - Prob. 58PSCh. 6.3 - Prob. 59PSCh. 6.3 - Prob. 60PSCh. 6.4 - Prob. 1PSCh. 6.4 - Prob. 2PSCh. 6.4 - Prob. 3PSCh. 6.4 - Prob. 4PSCh. 6.4 - Prob. 5PSCh. 6.4 - Prob. 6PSCh. 6.4 - Prob. 7PSCh. 6.4 - Prob. 8PSCh. 6.4 - Prob. 9PSCh. 6.4 - Prob. 10PSCh. 6.4 - Prob. 11PSCh. 6.4 - Prob. 12PSCh. 6.4 - Prob. 13PSCh. 6.4 - Prob. 14PSCh. 6.4 - Prob. 15PSCh. 6.4 - Prob. 16PSCh. 6.4 - Prob. 17PSCh. 6.4 - Prob. 18PSCh. 6.4 - Prob. 19PSCh. 6.4 - Prob. 20PSCh. 6.4 - Prob. 21PSCh. 6.4 - Prob. 22PSCh. 6.4 - Prob. 23PSCh. 6.4 - Prob. 24PSCh. 6.4 - Prob. 25PSCh. 6.4 - Prob. 26PSCh. 6.4 - Prob. 27PSCh. 6.4 - Prob. 28PSCh. 6.4 - Prob. 29PSCh. 6.4 - Prob. 30PSCh. 6.4 - Prob. 31PSCh. 6.4 - Prob. 32PSCh. 6.4 - Prob. 33PSCh. 6.4 - Prob. 34PSCh. 6.4 - Prob. 35PSCh. 6.4 - Prob. 36PSCh. 6.4 - Prob. 37PSCh. 6.4 - Prob. 38PSCh. 6.4 - Prob. 39PSCh. 6.4 - Prob. 40PSCh. 6.4 - Prob. 41PSCh. 6.4 - Prob. 42PSCh. 6.4 - Prob. 43PSCh. 6.4 - Prob. 44PSCh. 6.4 - Prob. 45PSCh. 6.4 - Prob. 46PSCh. 6.4 - Prob. 47PSCh. 6.4 - Prob. 48PSCh. 6.4 - Prob. 49PSCh. 6.4 - Prob. 50PSCh. 6.4 - Prob. 51PSCh. 6.4 - Prob. 52PSCh. 6.4 - Prob. 53PSCh. 6.4 - Prob. 54PSCh. 6.4 - Prob. 55PSCh. 6.4 - Prob. 56PSCh. 6.4 - Prob. 57PSCh. 6.4 - Prob. 58PSCh. 6.4 - Prob. 59PSCh. 6.4 - Prob. 60PSCh. 6.5 - Prob. 1PSCh. 6.5 - Prob. 2PSCh. 6.5 - Prob. 3PSCh. 6.5 - Prob. 4PSCh. 6.5 - Prob. 5PSCh. 6.5 - Prob. 6PSCh. 6.5 - Prob. 7PSCh. 6.5 - Prob. 8PSCh. 6.5 - Prob. 9PSCh. 6.5 - Prob. 10PSCh. 6.5 - Prob. 11PSCh. 6.5 - Prob. 12PSCh. 6.5 - Prob. 13PSCh. 6.5 - Prob. 14PSCh. 6.5 - Prob. 15PSCh. 6.5 - Prob. 16PSCh. 6.5 - Prob. 17PSCh. 6.5 - Prob. 18PSCh. 6.5 - Prob. 19PSCh. 6.5 - Prob. 20PSCh. 6.5 - Prob. 21PSCh. 6.5 - Prob. 22PSCh. 6.5 - Prob. 23PSCh. 6.5 - Prob. 24PSCh. 6.5 - Prob. 25PSCh. 6.5 - Prob. 26PSCh. 6.5 - Prob. 27PSCh. 6.5 - Prob. 28PSCh. 6.5 - Prob. 29PSCh. 6.5 - Prob. 30PSCh. 6.5 - Prob. 31PSCh. 6.5 - Prob. 32PSCh. 6.5 - Prob. 33PSCh. 6.5 - Prob. 34PSCh. 6.5 - Prob. 35PSCh. 6.5 - Prob. 36PSCh. 6.5 - Prob. 37PSCh. 6.5 - Prob. 38PSCh. 6.5 - Prob. 39PSCh. 6.5 - Prob. 40PSCh. 6.5 - Prob. 41PSCh. 6.5 - Prob. 42PSCh. 6.5 - Prob. 43PSCh. 6.5 - Prob. 44PSCh. 6.5 - Prob. 45PSCh. 6.5 - Prob. 46PSCh. 6.5 - Prob. 47PSCh. 6.5 - Prob. 48PSCh. 6.5 - Prob. 49PSCh. 6.5 - Prob. 50PSCh. 6.5 - Prob. 51PSCh. 6.5 - Prob. 52PSCh. 6.5 - Prob. 53PSCh. 6.5 - Prob. 54PSCh. 6.5 - Prob. 55PSCh. 6.5 - Prob. 56PSCh. 6.5 - Prob. 57PSCh. 6.5 - Prob. 58PSCh. 6.5 - Prob. 59PSCh. 6.5 - Prob. 60PSCh. 6.6 - Prob. 1PSCh. 6.6 - Prob. 2PSCh. 6.6 - Prob. 3PSCh. 6.6 - Prob. 4PSCh. 6.6 - Prob. 5PSCh. 6.6 - Prob. 6PSCh. 6.6 - Prob. 7PSCh. 6.6 - Prob. 8PSCh. 6.6 - Prob. 9PSCh. 6.6 - Prob. 10PSCh. 6.6 - Prob. 11PSCh. 6.6 - Prob. 12PSCh. 6.6 - Prob. 13PSCh. 6.6 - Prob. 14PSCh. 6.6 - Prob. 15PSCh. 6.6 - Prob. 16PSCh. 6.6 - Prob. 17PSCh. 6.6 - Prob. 18PSCh. 6.6 - Prob. 19PSCh. 6.6 - Prob. 20PSCh. 6.6 - Prob. 21PSCh. 6.6 - Prob. 22PSCh. 6.6 - Prob. 23PSCh. 6.6 - Prob. 24PSCh. 6.6 - Prob. 25PSCh. 6.6 - Prob. 26PSCh. 6.6 - Prob. 27PSCh. 6.6 - Prob. 28PSCh. 6.6 - Prob. 29PSCh. 6.6 - Prob. 30PSCh. 6.6 - Prob. 31PSCh. 6.6 - Prob. 32PSCh. 6.6 - Prob. 33PSCh. 6.6 - Prob. 34PSCh. 6.6 - Prob. 35PSCh. 6.6 - Prob. 36PSCh. 6.6 - Prob. 37PSCh. 6.6 - Prob. 38PSCh. 6.6 - Prob. 39PSCh. 6.6 - Prob. 40PSCh. 6.6 - Prob. 41PSCh. 6.6 - Prob. 42PSCh. 6.6 - Prob. 43PSCh. 6.6 - Prob. 44PSCh. 6.6 - Prob. 45PSCh. 6.6 - Prob. 46PSCh. 6.6 - Prob. 47PSCh. 6.6 - Prob. 48PSCh. 6.6 - Prob. 49PSCh. 6.6 - Prob. 50PSCh. 6.6 - Prob. 51PSCh. 6.6 - Prob. 52PSCh. 6.6 - Prob. 53PSCh. 6.6 - Prob. 54PSCh. 6.6 - Prob. 55PSCh. 6.6 - Prob. 56PSCh. 6.6 - Prob. 57PSCh. 6.6 - Prob. 58PSCh. 6.6 - Prob. 59PSCh. 6.6 - Prob. 60PSCh. 6 - Prob. 1PECh. 6 - Prob. 2PECh. 6 - Prob. 3PECh. 6 - Prob. 4PECh. 6 - Prob. 5PECh. 6 - Prob. 6PECh. 6 - Prob. 7PECh. 6 - Prob. 8PECh. 6 - Prob. 9PECh. 6 - Prob. 10PECh. 6 - Prob. 11PECh. 6 - Prob. 12PECh. 6 - Prob. 13PECh. 6 - Prob. 14PECh. 6 - Prob. 15PECh. 6 - Prob. 16PECh. 6 - Prob. 17PECh. 6 - Prob. 18PECh. 6 - Prob. 19PECh. 6 - Prob. 20PECh. 6 - Prob. 21PECh. 6 - Prob. 22PECh. 6 - Prob. 23PECh. 6 - Prob. 24PECh. 6 - Prob. 25PECh. 6 - Prob. 26PECh. 6 - Prob. 27PECh. 6 - Prob. 28PECh. 6 - Prob. 29PECh. 6 - Prob. 30PECh. 6 - Prob. 1SPCh. 6 - Prob. 2SPCh. 6 - Prob. 3SPCh. 6 - Prob. 4SPCh. 6 - Prob. 5SPCh. 6 - Prob. 6SPCh. 6 - Prob. 7SPCh. 6 - Prob. 8SPCh. 6 - Prob. 9SPCh. 6 - Prob. 10SPCh. 6 - Prob. 11SPCh. 6 - Prob. 12SPCh. 6 - Prob. 13SPCh. 6 - Prob. 14SPCh. 6 - Prob. 15SPCh. 6 - Prob. 16SPCh. 6 - Prob. 17SPCh. 6 - Prob. 18SPCh. 6 - Prob. 19SPCh. 6 - Prob. 20SPCh. 6 - Prob. 21SPCh. 6 - Prob. 22SPCh. 6 - Prob. 23SPCh. 6 - Prob. 24SPCh. 6 - Prob. 25SPCh. 6 - Prob. 26SPCh. 6 - Prob. 27SPCh. 6 - Prob. 28SPCh. 6 - Prob. 29SPCh. 6 - Prob. 30SPCh. 6 - Prob. 31SPCh. 6 - Prob. 32SPCh. 6 - Prob. 33SPCh. 6 - Prob. 34SPCh. 6 - Prob. 35SPCh. 6 - Prob. 36SPCh. 6 - Prob. 37SPCh. 6 - Prob. 38SPCh. 6 - Prob. 39SPCh. 6 - Prob. 40SPCh. 6 - Prob. 41SPCh. 6 - Prob. 42SPCh. 6 - Prob. 43SPCh. 6 - Prob. 44SPCh. 6 - Prob. 45SPCh. 6 - Prob. 46SPCh. 6 - Prob. 47SPCh. 6 - Prob. 48SPCh. 6 - Prob. 49SPCh. 6 - Prob. 50SPCh. 6 - Prob. 51SPCh. 6 - Prob. 52SPCh. 6 - Prob. 53SPCh. 6 - Prob. 54SPCh. 6 - Prob. 55SPCh. 6 - Prob. 56SPCh. 6 - Prob. 57SPCh. 6 - Prob. 58SPCh. 6 - Prob. 59SPCh. 6 - Prob. 60SPCh. 6 - Prob. 61SPCh. 6 - Prob. 62SPCh. 6 - Prob. 63SPCh. 6 - Prob. 64SPCh. 6 - Prob. 65SPCh. 6 - Prob. 66SPCh. 6 - Prob. 67SPCh. 6 - Prob. 68SPCh. 6 - Prob. 69SPCh. 6 - Prob. 70SPCh. 6 - Prob. 71SPCh. 6 - Prob. 72SPCh. 6 - Prob. 73SPCh. 6 - Prob. 74SPCh. 6 - Prob. 75SPCh. 6 - Prob. 76SPCh. 6 - Prob. 77SPCh. 6 - Prob. 78SPCh. 6 - Prob. 79SPCh. 6 - Prob. 80SPCh. 6 - Prob. 81SPCh. 6 - Prob. 82SPCh. 6 - Prob. 83SPCh. 6 - Prob. 84SPCh. 6 - Prob. 85SPCh. 6 - Prob. 86SPCh. 6 - Prob. 87SPCh. 6 - Prob. 88SPCh. 6 - Prob. 89SPCh. 6 - Prob. 90SPCh. 6 - Prob. 91SPCh. 6 - Prob. 92SPCh. 6 - Prob. 93SPCh. 6 - Prob. 94SPCh. 6 - Prob. 95SPCh. 6 - Prob. 96SPCh. 6 - Prob. 97SPCh. 6 - Prob. 98SPCh. 6 - Prob. 99SP
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- Find the exact volume of the solid that results when the rectangular region with vertices at 0, 0, 6, 0, 6, 4 and 0, 4 is revolved about the a x-axis b y-axisarrow_forward2. The figure shows the curves of the equations: y = 2x², y = 4x - 2, and y = 8. Find the volume of the solid obtained by rotating the region bounded by the three curves about the y-axis. (Just express it as an integral formula, and its value does not to be calculated.) (a) Use the method of washers (b) Use the method of cylindrical shells (Hint: You need divide the region into two parts first.)arrow_forward4. Sketch the graphs of y = 4 + (x)", x = 9 and y = 4. Shade the region bounded by the curves. Find the volume of the solid that is formed by revolving the region shaded about (a) the x-axis (b) the y-axis.arrow_forward
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