volume between two graphs over a region of the plane
(with partial cross-section planes of double integration)Stewart 8e 15.1.45. Find the volume between planes 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 and the surface (paraboloid) 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.This describes the region of space as
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 while 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjBGJ0Y5LUY2Ni1RIy4uRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZOUSYwLjBlbUYnLUZQNiRRIjRGJ0Y5Rjk=.
Its volume is the double integral over a rectangle in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=-LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= plane of the difference of the two functions of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.
We plot the bottom and top of this region of space:LUkncGxvdDNkRzYiNiY3JCIiIiwoIiIjRicqJEkieEdGJEYpRicqJCwmSSJ5R0YkRichIiNGJ0YpRicvRis7ISIiRicvRi47IiIhIiIlL0klYXhlc0dGJEkmYm94ZWRHRiQ=Guestimating the volume, hmm, area of base is 2x4 = 8, height is 6 (notice bottom tickmark is 1), volume of cube 48, it fills less than half the volume looking from the side, so somewhat under 24...how does that compare with the exact answer?better plot for fun (uses/needs parametrized surfaces!)We show the four sides as well as the top and bottom.QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIiPkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwoIiIjIiIiKiQ5JEYtRi4qJCwmOSVGLiEiI0YuRi1GLkYkRiRGJA==LUkoZGlzcGxheUc2IjYoLUkncGxvdDNkRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiY3JCIiIiwoIiIjRi0qJEkieEdGJEYvRi0qJCwmSSJ5R0YkRi0hIiNGLUYvRi0vRjE7ISIiRi0vRjQ7IiIhIiIlL0klYXhlc0dGJEkmYm94ZWRHRiQtRic2JTclRjFGO0kiekdGJEY2L0ZDO0YtLUkiZkdGJDYkRjFGOy1GJzYlNyVGMUY8RkNGNi9GQztGLS1GRzYkRjFGPC1GJzYlNyVGOEY0RkNGOS9GQztGLS1GRzYkRjhGNC1GJzYlNyVGLUY0RkNGOS9GQztGLS1GRzYkRi1GNC9JKHNjYWxpbmdHRiRJLGNvbnN0cmFpbmVkR0YkWhile we are at it we can put in the partial integration cross-sections, like the partial derivative cross-sections:LUkoZGlzcGxheUc2JCUqcHJvdGVjdGVkRy9JK21vZHVsZW5hbWVHNiJJJnBsb3RzRzYkRiVJKF9zeXNsaWJHRig2Ji1JJ3Bsb3QzZEdGKjYmNyQiIiIsKCIiI0YxKiRJInhHRihGM0YxKiQsJkkieUdGKEYxISIjRjFGM0YxL0Y1OyEiIkYxL0Y4OyIiISIiJS9JJWF4ZXNHRihJJmJveGVkR0YoLUYuNiU3JUY1RjNJInpHRihGOi9GRztGMS1JImZHRig2JEY1RjMtRi42JTclRj9GOEZHRj0vRkc7RjEtRks2JEY/RjgvSShzY2FsaW5nR0YoSSxjb25zdHJhaW5lZEdGKA==QyctSSRpbnRHNiI2JC1GJDYkLUkiK0clKnByb3RlY3RlZEc2JCwoIiIjIiIiKiRJInhHRiVGLkYvKiQsJkkieUdGJUYvISIjRi9GLkYvISIiL0Y0OyIiISIiJS9GMTtGNkYvRi8tSSZldmFsZkdGKzYjSSIlR0YlRi8vSSxmcmFjdGlvbmJveEdGJSomRkBGLy1JIi9HNiRGK0koX3N5c2xpYkdGJTYjLCQtSSIqR0YrNiRGLkY6IiInRi8=By hand we would do the following evaluation steps:Qy0tSSIrRyUqcHJvdGVjdGVkRzYkLCgiIiMiIiIqJEkieEc2IkYoRikqJCwmSSJ5R0YsRikhIiNGKUYoRikhIiJGKS1JJGludEc2JEYlSShfc3lzbGliR0YsNiRJIiVHRixGL0YpLCYtSSVldmFsR0YlNiRGNy9GLyIiJUYpLUY6NiRGNy9GLyIiIUYxRiktRjM2JEY3RitGKSwmLUY6NiRGNy9GK0YpRiktRjo2JEY3L0YrRjFGMUYpLUkmZXZhbGZHRiU2I0Y3JSFH