MTH 251 Lecture Notes - Lecture 36: Cartesian Coordinate System

Show all your work for full credit. The textbook, lecture notes, and elec- tronic devices with symbolic computing capacity are NOT allowed 1. Evaluate cos(x + 2y)dA, whe 2. Evaluate the following double integral Jo Jz 3. Evaluate the double integrals by converting them to polar coordinates. 2心ー昏 4. Evaluate the double integrals by converting them to polar coordinates. 5. Evaluate the triple integrals dxdydz, 2 drdzdy, 3. rdzdrd9. 6. Find the volume between the graph of the function f(r, y) - zy and the rect- angle R = {(z, y)1-2 3and-1 7. Evaluate the following double integrals y 5) 0 z+2 3. 2ded 8. Let S be the triangular region with vertices (0,0), (-1, 1), and (1, 1). (Sketch the region first!) . Find the centroid ofS . If the density at each point in S is proportional to the point's distance from the y-axis, find the mass of S.

Parabolic mirrors have the nice focusing property that all rays coming in parallel to their axis of symmetry are focused to the same point. Suppose that we have a parabolic mirror surface that is implicitly defined by the equation F(x, y, z) = 0, where F(x, y, z) = x2 + y2 - z. Use the gradient of F(x, y, z) to find a vector normal to the surface at the point (1.2,5). What is the standard equation for the plane tangent, T. to our paraboloid at (1, 2, 5)? What is the normal vector to T? Consider a point on a surface with a local normal n, and a ray of light that reflects off the surface at that point. Let the direction of the incoming and reflected rays be win and wout, respectively. The formula for finding the direction of the reflected ray, given the incoming ray and the local normal, wout = win - 2 (win · n)/(n · n)n. Draw a picture and explain why this gives us vectors where the angle of incidence equals the angle of reflection. Consider an incoming ray that is parallel to the z-axis so that a vector in the direction of this ray is win = - k. What is wout if win intersects the mirror at (1, 2, 5)? What is the parameterization for the line, L, that intersects the surface at the point (1, 2, 5) and is in the direction of wout? The point where L intersects the z-axis is the focal point of the parabolic mirror. Determine the coordinates of the focal point for our mirror.

Parabolic mirrors have the nice focusing property that all rays coining in parallel to their axis of symmetry are focused to the same point. Suppose that we have a parabolic mirror surface that is implicitly defined by the equation F(x. y. z) = 0, where F(x,y,z) = x 2 + y 2 - z. Consider a point on a surface with a local normal n, and a ray of light that reflects off the surface at that point. Let the direction of the incoming and reflected rays be w in and w out, respectively. The formula for finding the direction of the reflected ray, given the incoming ray and the local normal. w out = w in - 2 (w in n)/(n n) n. Draw a picture and explain why this gives us vectors where the angle of incidence equals the angle of reflection. Consider an incoming ray that is parallel to the z-axis so that a vector in the direction of this ray is w in = - k. What is W out if w in intersects the mirror at (1,2,5)? What is the parameterization for the line, L, that intersects the surface at the point (1,2,5) and is in the direction of w out? The point where L intersects the z-axis is the focal point of the parabolic mirror. Determine the coordinates of the focal point for our mirror.