Applications of Double Integrals: Surface Area Calc IV Lab LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjhRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdGPC8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0Y8LyUqc3ltbWV0cmljR0Y8LyUobGFyZ2VvcEdGPC8lLm1vdmFibGVsaW1pdHNHRjwvJSdhY2NlbnRHRjwvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJ0Y6Rj0=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 JSFH

The surface area of f, a function of two variables, over a region R can be computed with the following double integral: 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 Maple's MultivariateCalculus package has the command SurfaceArea that will perform this calculation. As an example we calculate the surface area of the function f(x, y) = x^2+y* over the rectangular region 0 \342\211\244 x \342\211\244 2, 0 \342\211\244 y \342\211\244 3 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= In this form we ask for the output integral form and then ask for the value of that integral: 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 In this form, we just cut to the chase and get the exact, and then approximate, answer. 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 JSFH If we add the option "output = plot", we see a plot of the surface over the desired region of integration. 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 JSFH

Example: Calculate the surface area of the paraboloid LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYmUSJ6RicvJSVzaXplR1EjMTJGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEiPUYnRjRGOi8lJmZlbmNlR0Y5LyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtRiM2KS1JI21uR0YkNiVRIjRGJ0Y0RjotRj42LlEoJm1pbnVzO0YnRjRGOkZBRkNGRUZHRklGS0ZNL0ZQUSwwLjIyMjIyMjJlbUYnL0ZTRmhuLUYjNiQtSSVtc3VwR0YkNiUtRiw2JlEieEYnRjRGN0Y6LUZXNiVRIjJGJ0Y0RjovJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjpGWi1GIzYkLUZdbzYlLUYsNiZRInlGJ0Y0RjdGOkZib0Zlb0Y6RitGOkYrRjpGK0Y6 over the rectangular region R with vertices (0,0), (2,0), (2,3), (0,3). 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 When we ask for the answer, even Maple can't find an exact answer (no antiderivative for the outer integral) -- so we must ask for an approximation: 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 Here we show the steps: 1. Evaluate the partial derivatives: 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 2. Set up the double integral; 3) compute the integral (attempt to); 4) approximate 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

Exercise 1: Compute the surface area for the same paraboloid over each of the following regions (you do not need to plot) (a) R is the region bounded by the x-axis and the lines LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYmUSJ4RicvJSVzaXplR1EjMTJGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEiPUYnRjRGOi8lJmZlbmNlR0Y5LyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYlUSIzRidGNEY6RjpGK0Y6 and 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= (b) R is the region in the first quadrant bounded by the y-axis, the line LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2I1EhRictRiM2Jy1GLDYmUSJ5RicvJSdmYW1pbHlHUSZBcmlhbEYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSNtb0dGJDYuUSI9RidGNEY6LyUmZmVuY2VHRjkvJSpzZXBhcmF0b3JHRjkvJSlzdHJldGNoeUdGOS8lKnN5bW1ldHJpY0dGOS8lKGxhcmdlb3BHRjkvJS5tb3ZhYmxlbGltaXRzR0Y5LyUnYWNjZW50R0Y5LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUS1JI21uR0YkNiVRIjRGJ0Y0RjpGNEY6RitGNEY6 and 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= (c) R is the region bounded by 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. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Exercise 2: Complete Exercise 22, Section 14.5 Exercise 3: Complete Exercise 39, section 14.5. Below are the commands to plot the surfaces representing the floor and the ceiling. 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 Part a) The volume: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH Part b) Surface Area of the Ceiling: JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=