Problem 1QCE: The surface area of a surface of the form z=fx,y over a region R in the xy-plane is given by S=RdA Problem 2QCE: Consider the surface represented parametrically by x=1uy=1ucosv0u1,0v2z=1usinv (a) Describe the... Problem 3QCE: If ru,v=1ui+1ucosvj+1usinvk then ru=andrv= Problem 4QCE: If ru,v=1ui+1ucosvj+1usinvk the principal unit normal to the graph of r at the point where... Problem 5QCE: Suppose is a parametric surface with vector equation ru,v=xu,vi+u,vj+zu,vk If has no... Problem 1ES: Express the area of the given surface as an iterated double integral, and then find the surface... Problem 2ES Problem 3ES: Express the area of the given surface as an iterated double integral, and then find the surface... Problem 4ES Problem 5ES Problem 6ES Problem 7ES Problem 8ES: Express the area of the given surface as an iterated double integral in polar coordinates, and then... Problem 9ES: Express the area of the given surface as an iterated double integral in polar coordinates, and then... Problem 10ES: Express the area of the given surface as an iterated double integral in polar coordinates, and then... Problem 11ES Problem 12ES: Sketch the parametric surface. ax=u,y=v,z=u2+v2bx=u,y=u2+v2,z=vcx=u2+v2,y=u,z=v Problem 13ES Problem 14ES Problem 15ES Problem 16ES Problem 17ES: Find parametric equations for the surface generated by revolving the curve about the
Problem 18ES: Find parametric equations for the surface generated by revolving the curve yex=0 about the x-axis. Problem 19ES Problem 20ES Problem 21ES: Find a parametric representation of the surface in terms of the parameters r and where are the... Problem 22ES Problem 23ES: Find a parametric representation of the surface in terms of the parameters r and where are the... Problem 24ES Problem 25ES: Find a parametric representation of the cone z=3x2+3y2 in terms of parameters and , where ,, are... Problem 26ES Problem 27ES: Eliminate the parameters to obtain an equation in rectangular coordinates, and describe the surface.... Problem 28ES Problem 29ES Problem 30ES Problem 31ES: Eliminate the parameters to obtain an equation in rectangular coordinates, and describe the surface.... Problem 32ES Problem 33ES: The accompanying figure shows the graphs of two parametric representations of the cone z=x2+y2 for... Problem 34ES: The accompanying figure shows the graphs of two parametric representations of the paraboloid... Problem 35ES: In each part, the figure shows a portion of the parametric surface x=3cosv,y=u,z=3sinv. Find... Problem 36ES: In each part, the figure shows a portion of the parametric surface x=3cosv,y=3sinv,z=u. Find... Problem 37ES: In each part, the figure shows a hemisphere that is a portion of the sphere x=sincos,y=sinsin,z=cos.... Problem 38ES: In each part, the figure shows a portion of the sphere x=sincos,y=sinsin,z=cos. Find restrictions on... Problem 39ES: Find an equation of the tangent plane to the parametric surface at the stated point.... Problem 40ES Problem 41ES: Find an equation of the tangent plane to the parametric surface at the stated point.... Problem 42ES Problem 43ES: Find an equation of the tangent plane to the parametric surface at the stated point.... Problem 44ES Problem 45ES: Find the area of the given surface. The portion of the paraboloid ru,v=ucosvi+usinvj+u2k for which... Problem 46ES: Find the area of the given surface. The portion of the cone ru,v=ucosvi+usinvj+uk for which... Problem 47ES: Determine whether the statement is true or false. Explain your answer. If f has continuous first... Problem 48ES: Determine whether the statement is true or false. Explain your answer. Suppose that z=fx,y has... Problem 49ES: Determine whether the statement is true or false. Explain your answer. If ru,v=xu,vi+yu,vj+zu,vk... Problem 50ES: Determine whether the statement is true or false. Explain your answer. For the function fx,y=ax+by,... Problem 51ES: Use parametric equations to derive the formula for the surface area of a sphere of radius a. Problem 52ES: Use parametric equations to derive the formula for the lateral surface area of a right circular... Problem 53ES: The portion of the surface z=hax2+y2a,h0 between the xy-plane and the plane z=h is a right circular... Problem 54ES: The accompanying figure shows the torus that is generated by revolving the circle xa2+z2=b20ba in... Problem 55ES Problem 56ES: Use a CAS to graph the helicoid x=ucosv,y=usinv,z=v for 0u5and0v4 (see the accompanying figure on... Problem 57ES: Use a CAS to graph the pseudosphere x=cosusinvy=sinusinvz=cosv+lntanv2 for 0u2,0v (see the... Problem 58ES: (a) Find parametric equations for the surface of revolution that is generated by revolving the curve... Problem 59ES: The parametric equations in these exercises represent a quadric surface for positive values of a, b,... Problem 60ES: The parametric equations in these exercises represent a quadric surface for positive values of a, b,... Problem 61ES: The parametric equations in these exercises represent a quadric surface for positive values of a, b,... Problem 62ES format_list_bulleted