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# Calculus 3 MATH 311

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This 13 page Class Notes was uploaded by Vernie McCullough on Thursday October 15, 2015. The Class Notes belongs to MATH 311 at Millersville University of Pennsylvania taught by J. Buchanan in Fall. Since its upload, it has received 14 views. For similar materials see /class/223528/math-311-millersville-university-of-pennsylvania in Mathematics (M) at Millersville University of Pennsylvania.

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Date Created: 10/15/15

Surface Area MATH 311 Calculus II J Robert Buchanan Department of Mathematics Fall 2007 Review Previously we learned a formula for determining the surface area for a solid of revolution Suppose y fX 2 O for a g X g b is revolved around the Xaxis The surface area of the solid of revolution is s 27139 b mi 1 rx2 dx Review Previously we learned a formula for determining the surface area for a solid of revolution Suppose y fX 2 O for a g X g b is revolved around the Xaxis The surface area of the solid of revolution is s 27139 b mi 1 rx2 dx Today we learn how to find the surface area of a surface over a bounded region R using a double integral Riemann Sum Approach Let P be an inner partition of H and place a tangent plane on the surface at the corner closest to the origin of each area element in P r Tangent Planes The sum of the areas of the tangent planes will approximate the surface area of the function over H r l A Single Tangent Plane 7 7 If Xkyk is the corner of Hk closest to the origin then tangent plane touches the surface at XMYM XkJkD A vector tangent to the surface and parallel to the xzplane is a lt1707 fxXk7YKgt A vector tangent to the surface and parallel to the yzplane is b 0717 fyXk7YKgt Area of a Single Tangent Plan 7 The tangent plane element is a parallelogram whose area is lAxka X Ay bll lla X bHAXkAyk Hlt1707fxxk7kgt X lt0717fyXk7kgt lAAk lrxrxmmz Mam 1AAk Ark rxrxmnz 0mm 1AAk Formula The Riemann sum approximating the surface area for fX y over H is 8 z iATk k1 Formula The Riemann sum approximating the surface area for fX y over H is n S x Z ATk k1 Consequently n S HFIJHnlogATk n wy mo 2 0mm fyXk7YIlt2 1AAk k1 M xrxw my 1dA Example Find the surface area of the part of the surface 2 X2 2y that lies above the triangular region in the xyplane with vertices at 00 10 and 1 1 5 it away 14511quot u 401iquot quotlilting 7 1 Find the surface area of the part of the paraboloid z X2 y2 that lies below the plane where z 9 V WW Find the surface area of the part of the surface 2 4 7 X2 above 2 O and between y O and y 4 Egyn v quot 4 I v I if Homework 0 Read Section 134 0 Pages 10711072 1 25 odd

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