. Differential and integral calculus, an introductory course for colleges and engineering schools. t, there is seldom any advantage in using a double integralrather than a single integral to calculate a plane area. Problem. Express as a double integral in polar coordinates, p, 6, thearea of a sector bounded by a curve and two radii vectores. Suggestion. Divide the sector into curvilinear quadrilaterals by radiiand concentric circles. 216. The Area of a Surface Expressed as a Double Integral. The area of a curved surface can be expressed as a double in-tegral extended over a plane region. To es
. Differential and integral calculus, an introductory course for colleges and engineering schools. t, there is seldom any advantage in using a double integralrather than a single integral to calculate a plane area. Problem. Express as a double integral in polar coordinates, p, 6, thearea of a sector bounded by a curve and two radii vectores. Suggestion. Divide the sector into curvilinear quadrilaterals by radiiand concentric circles. 216. The Area of a Surface Expressed as a Double Integral. The area of a curved surface can be expressed as a double in-tegral extended over a plane region. To establish the formulawe shall have need of the following Lemma. Let A be any area in a plane M, and A its (orthog-onal) projection upon any other plane 6 be the angle between the planes. Then A = A cos 0. To prove this, we first cover A with a net-work of small rectangles by drawing linesparallel and perpendicular to L, the lineof intersection of M and M. This net-work projects into a similar network of rectangles covering a be one of the rectangles of A, and a its projection in 216 MULTIPLE INTEGRALS 329 It is easily proved that a = acos0, and from this it follows thatlim Va= cosfllim^a, or A = Acosd. Q. E. D. Now let the equation of the curved surface be f(x, y, z) = 0,and let S be a portion of this surface bounded by a closed contour seek a formula for the area of S. Let S be projected into a region S of the xy-plsme. We assumethat the perpendiculars upon the x?/-plane which project S intoS meet S in but one point; that is, weassume z to be a single-valued function ofx and y throughout S.* Circumscribeabout S a polyhedral surface P, by con-structing tangent planes to S. As thenumber of tangent planes is indefinitelyincreased, the number of faces of P isindefinitely increased, and the area of Papproaches as a limit the area of $.f Rep-resenting the faces of P by APi, AP2, • • • ,it is fairly obvious that
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
