Question

Please help with 1-10 and please show all work thanks.

Show all of your work neatly, and express solutions as exact answers unless otherwise requested. No credit will be given to solutions that have no work shown! BOX or CIRCLE your final answer. 1. Sketch a graph and shade the area of the region bounded by the following equations. Set up an integral that would give this area. 2x + y2 = 6 and y=x+1 2. Sketch a graph and shade the region bounded by f(x) = Vx, the y-axis, and the line y = 1. Set up TWO integrals that could be used to find the volume of the solid obtained by rotating this region about the y-axis, but DO NOT evaluate either integral. A. Using the Disk Method: B. Using Cylindrical Shells: 3. Sketch a graph and shade the region bounded by y = x, y = e*, x = 1, and the y-axis. Set up an integral that could be used to find the volume of the solid obtained by rotating this region about the line x = -1. DO NOT evaluate the integral 4. Sketch a graph and shade the region bounded by y = x and y = 1. Set up an integral to find the volume of the solid obtained by rotating this region about the line y = -1. DO NOT evaluate the integral. 5. Sketch a graph and shade the region bounded by x + y2 = 2 and x = 0. Set up an integral that could be used to find the volume of the solid obtained by rotating this bounded region about the line x = -2. DO NOT evaluate the integral. 6. Find the length of the curve defined by f(x) from x = 0 to x = 1. Evaluate, show work, and give an exact solution. f(x) = V1 - x2 + sin-x

Answer 1

wort to (1,2) 2=y-le I Soph Hese integration y will be easy. Right f = a= 6-42 left f = 2 = y1 intersection point 2 = 6- -5, -4) =) 6-92 = 29-2 =) ytzy - 8 = 0 o; y = -4,2 Arce between were OW = pright ft – left f) dy y=a S 42 – (4-0) ay (2) rotated about y-axis ya! (161) DISK Method. inner radius , r=0 outes racing , R = x = y² -0.5 0.5 х so, volume, v= II Ske-82) dy

llogg?-0) dy = r syv dy (b) SHELL / CYLINDRICAL METHOD Considy a rectangulal Stoip mu parallel to y = 1 the axis of rotation y-axis. -0.54 So, radius, r(n) = a height MM, h(a) = (1-1) -0.5 0.5 ; v = 211 fras, hem da この 271 In) da 2 = -1 (3) SHELL METHOD r(n) = (1+1) h(n) = 19-21) : V= 20 from him) dan r() = (n+I |(1,1) Ny=n | = 20 % (1+a) (en–a) da -0.5 0.5