Applications of Double Integrals: Surface Area Calc IV Lab LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2Iy1GLDYlUShyZXN0YXJ0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4yNzc3Nzc4ZW1GJw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JUYrLUYjNiUtRiw2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y6USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZTLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEmcGxvdHNGJ0Y2RjlGQEYrRis= LUkld2l0aEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjJkkoU3R1ZGVudEdGJzYjSTVNdWx0aXZhcmlhdGVDYWxjdWx1c0dGJw== 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\244y\342\211\2443 In this form we ask for the output integral form and then ask for the value of that integral: QyUtSSxTdXJmYWNlQXJlYUc2IjYmLCYqJEkieEdGJSIiIyIiIkkieUdGJUYrL0YpOyIiIUYqL0YsO0YvIiIkL0knb3V0cHV0R0YlSSlpbnRlZ3JhbEdGJUYrLUkmdmFsdWVHSShfc3lzbGliR0YlNiNJIiVHRiU= In this form, we just cut to the chase and get the exact, and then approximate, answer. QyUtSSxTdXJmYWNlQXJlYUc2IjYlLCYqJEkieEdGJSIiIyIiIkkieUdGJUYrL0YpOyIiIUYqL0YsO0YvIiIkRistSSZldmFsZkclKnByb3RlY3RlZEc2I0kiJUdGJQ== JSFH If we add the option "output = plot", we see a plot of the surface over the desired region of integration. LUksU3VyZmFjZUFyZWFHNiI2JiwmKiRJInhHRiQiIiMiIiJJInlHRiRGKi9GKDsiIiFGKS9GKztGLiIiJC9JJ291dHB1dEdGJEklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJA== JSFH

Example: Calculate the surface area of the paraboloid 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 over the rectangular region R with vertices (0,0), (2,0), (2,3), (0,3). PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwoIiIlIiIiKiQ5JCIiIyEiIiokOSVGMUYyRiRGJEYk LUksU3VyZmFjZUFyZWFHNiI2Ji1JImZHRiQ2JEkieEdGJEkieUdGJC9GKTsiIiEiIiMvRio7Ri0iIiQvSSdvdXRwdXRHRiRJKWludGVncmFsR0Yk LUksU3VyZmFjZUFyZWFHNiI2Ji1JImZHRiQ2JEkieEdGJEkieUdGJC9GKTsiIiEiIiMvRio7Ri0iIiQvSSdvdXRwdXRHRiRJJXBsb3RHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ= 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: QyUtSSxTdXJmYWNlQXJlYUc2IjYlLUkiZkdGJTYkSSJ4R0YlSSJ5R0YlL0YqOyIiISIiIy9GKztGLiIiJCIiIi1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjSSIlR0Yl Here we show the steps: 1. Evaluate the partial derivatives: QyQ+SSNmeEc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJmR0YlNiRJInhHRiVJInlHRiVGLSIiIg==PkkjZnlHNiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiZkdGJDYkSSJ4R0YkSSJ5R0YkRi0= 2. Set up the double integral; 3) compute the integral (attempt to); 4) approximate QyctSSlNdWx0aUludEc2IjYmLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCgiIiJGLiokSSNmeEdGJSIiI0YuKiRJI2Z5R0YlRjFGLi9JInhHRiU7IiIhRjEvSSJ5R0YlO0Y3IiIkL0knb3V0cHV0R0YlSSlpbnRlZ3JhbEdGJUYuLUkmdmFsdWVHRis2I0kiJUdGJUYuLUkmZXZhbGZHRipGQQ==

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2I1EhRictRiM2JS1GLDYmUSJ4RicvJSVzaXplR1EjMTJGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LlEiPUYnRjRGOi8lJmZlbmNlR0Y5LyUqc2VwYXJhdG9yR0Y5LyUpc3RyZXRjaHlHRjkvJSpzeW1tZXRyaWNHRjkvJShsYXJnZW9wR0Y5LyUubW92YWJsZWxpbWl0c0dGOS8lJ2FjY2VudEdGOS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYlUSIzRidGNEY6Ris= and 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. QyQ+SSJmRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgiIiUiIiIqJDkkIiIjISIiKiQ5JUYyRjNGJUYlRiVGLw==QyQ+SSNmeEc2Ii1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJmR0YlNiRJInhHRiVJInlHRiVGLSIiIg==PkkjZnlHNiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiZkdGJDYkSSJ4R0YkSSJ5R0YkRi0= QyctSSlNdWx0aUludEc2IjYmLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCgiIiJGLiokSSNmeEdGJSIiI0YuKiRJI2Z5R0YlRjFGLi9JInlHRiU7LUkiKkdGKjYkLCRJInhHRiVGMSNGLiIiJEYxL0Y7OyIiIUY9L0knb3V0cHV0R0YlSSlpbnRlZ3JhbEdGJUYuLUkmdmFsdWVHRis2I0kiJUdGJUYuLUkmZXZhbGZHRipGRg== (b) R is the region in the first quadrant bounded by the y-axis, the line 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 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2I1EhRictRiM2Ky1GLDYlUSJ5RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0Y3LyUmZmVuY2VHRjYvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGNi8lKnN5bW1ldHJpY0dGNi8lKGxhcmdlb3BHRjYvJS5tb3ZhYmxlbGltaXRzR0Y2LyUnYWNjZW50R0Y2LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1GIzYoLUklbXN1cEdGJDYlLUYsNiVRInhGJ0Y0RjctSSNtbkdGJDYkUSIyRidGNy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvJSdmYW1pbHlHUSZBcmlhbEYnLyUlc2l6ZUdRIjlGJy8lK2JhY2tncm91bmRHUShbMCwwLDBdRicvJTBmb250X3N0eWxlX25hbWVHUSdOb3JtYWxGJ0Y3RitGam5GXW9GYG9GY29GN0YrRmpuRl1vRmBvRmNvRjc=. QyctSSlNdWx0aUludEc2IjYmLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCgiIiJGLiokSSNmeEdGJSIiI0YuKiRJI2Z5R0YlRjFGLi9JInlHRiU7KiRJInhHRiVGMSIiJS9GODsiIiFGMS9JJ291dHB1dEdGJUkpaW50ZWdyYWxHRiVGLi1JJnZhbHVlR0YrNiNJIiVHRiVGLi1JJmV2YWxmR0YqRkI= (c) R is the region bounded by 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. QyctSSlNdWx0aUludEc2IjYmLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCgiIiJGLiokSSNmeEdGJSIiI0YuKiRJI2Z5R0YlRjFGLi9JInlHRiU7LCQtRig2IywmIiIlRi4qJEkieEdGJUYxISIiRj5GOC9GPTshIiNGMS9JJ291dHB1dEdGJUkpaW50ZWdyYWxHRiVGLi1JJnZhbHVlR0YrNiNJIiVHRiVGLi1JJmV2YWxmR0YqRkc= Exercise 22, Section 14.5 QyctSSlNdWx0aUludEc2IjYmKiYtSSVzcXJ0RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJiIiIkYvKiRJInJHRiUiIiMiIiVGL0YxRi8vRjE7IiIhRjMvSSZ0aGV0YUdGJTtGNiwkSSNQaUdGK0YyL0knb3V0cHV0R0YlSSlpbnRlZ3JhbEdGJUYvLUkmdmFsdWVHRiw2I0kiJUdGJUYvLUkpc2ltcGxpZnlHRipGQQ== Exercise 3: Complete exercise 39, section 14.5 Part a) The volume: PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwmIiM/IiIiKiY5JUYuOSRGLiNGLiIkKyJGJEYkRiQ= PkkiZ0c2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwmOSQjIiIiIiImOSVGLkYkRiRGJA== QyctSSlNdWx0aUludEc2IjYmLCYtSSJmR0YlNiRJInhHRiVJInlHRiUiIiItSSJnR0YlRiohIiIvRis7IiIhLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCYtSSJeR0Y3NiQiI10iIiNGLSokRixGP0YwL0YsO0YzRj4vSSdvdXRwdXRHRiVJKWludGVncmFsR0YlRi0tSSZ2YWx1ZUdGODYjSSIlR0YlRi0tSSZldmFsZkdGN0ZI JSFH Surface Area of the Ceiling: JSFH PkkiZkc2ImYqNiRJInhHRiRJInlHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwmIiM/IiIiKiY5JUYuOSRGLiNGLiIkKyJGJEYkRiQ= PkkjZnhHNiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiZkdGJDYkSSJ4R0YkSSJ5R0YkRiw= PkkjZnlHNiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkiZkdGJDYkSSJ4R0YkSSJ5R0YkRi0= QyctSSlNdWx0aUludEc2IjYmLUklc3FydEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCgiIiJGLiokSSNmeEdGJSIiI0YuKiRJI2Z5R0YlRjFGLi9JInhHRiU7IiIhLUYoNiMsJi1JIl5HRio2JCIjXUYxRi4qJEkieUdGJUYxISIiL0ZAO0Y3Rj4vSSdvdXRwdXRHRiVJKWludGVncmFsR0YlRi4tSSZ2YWx1ZUdGKzYjSSIlR0YlRi4tSSZldmFsZkdGKkZJ 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