Advanced Calculus fi..

248 Advanced Calculus, Fifth Edition !i:3 iwrAccordingly, by (4.86),ds2 = E du2 + 2~ du dv + G dv2. (4.89)This shows the significance of E, F, G for the geometry on the surface.PROBLEMS 41. Find the length of the circumference of a circlea) using the parametric representationIb) using the parametric representation1-t2x=u-y =a-2t1 + t2' 1+t2'2. Find the area of the surface of a spherea) using the equationz = Ja2--,b) using the parametric equations a' c ) 13. Find the surface area of the surface having the given parametric equations: 1a) x = (b+acosv)co~u, y = (b+acosv)sinu, z = asinv, 0 5 u p 217, 0 p v 52x, where a and b are constants, 0 < a < b (torus).b) x = u cos v, y = u sin v, z = u2 sin2v, 0 5 u p 1, 0 p v 5 x/2 (portion ofsaddle surface z = 2xy).4. Using the parametric equations of Problem 2(b), set up a double integral for the area of aportion of the earth's surface bounded by two parallels of latitude and two meridians oflongitude. Apply this to find the area of the United States of America, approximating thisby the "rectangle" between parallels 30°N and 47"N and meridians 75"W and 122"W.Take the radius of the earth to be 4000 miles.5. Let a parallelogranl be given in space whose sides represent the vectors a and b. Let c bea unit vector perpendicular to a plane C.a) 'Show that a x b . c equals plus or minus the area of the projection of the parallelogramon C.b) Show that this can also be written as S cos y, where S is the area of the parallelogramand y is the angle between a x b and c.C) Show that one hass = Jw,where S,, , S,, , S,, are the areas of the projections of the parallelogram on the yz-plane,a-plane, q-plane.6. A surface of revolution is obtained by rotating a curve z = f (x), y = 0 in the xz-planeabout the z-axis.a) Show that this surface has the equation z = f (r) in cylindrical coordinates.b) Show that the area of the surface is1