7. [-/0.25 Points] DETAILS SCALCET9 8.3.045. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Use the Theorem of Pappus to find the volume of the given solid. A cone with height h and base radius V = Need Help? Read It Watch It 8. [-/0.25 Points] DETAILS SCALCET9 8.XP.3.018. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Calculate the moments M and M and the center of mass of a lamina with the given density and shape. p = 2 (6, 5) My = My = ( x, y ) = Need Help? Read It Watch It 9. [-/0.27 Points] DETAILS SCALCET9 8.XP.3.011.MI. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Visually estimate the location of the centroid of the region shown. Then find the exact coordinates of the centroid. y = 49 - x2, y = 0 60 50 40 30 20 10 -6 -4 -2 2 4 6 ( x , v ) = Need Help? Read It Master ItClick here to view the transcript. center of mass 3 EXAMPLE Find the moments and center of mass of the system of objects that have masses 3, 3, and 6 at the point (-1, 1), (2, -1), and (3, 2), respectively. SOLUTION We use the following equations to compute the moments: My = 3(-1) + 3(2) + 6(3) O My = 3(1) + 3(-1) + 6(2) Since m = 3 + 3 + 6 = , we use these equations to obtain the following. X = My = m y = _ MX = [ m Thus the center of mass is (x, y) = (See the figure.) Need Help? Read It 2. [-/0.25 Points] DETAILS SCALCET9 8.3.AE.007. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Example Video Example .() A torus is formed by rotating a circle of radius r about a line in the plane of the circle that is a distance R (> r) from the center of the circle. Find the volume of the torus. Solution The circle has area A = By the symmetry principle, its centroid is its center and so the distance traveled by the centroid during a rotation is d= . Therefore, by the Theorem of Pappus, find the volume of the torus. V = Ad = (2R) Need Help? Read It 3. [-/0.25 Points] DETAILS SCALCET9 8.3.021.MI. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER Point-masses m, are located on the x-axis as shown. Find the moment M of the system about the origin and the center of mass x. m1 = 4 m2 = 12 5 10 15 20 25 30 35 40 M X Need Help? Read It Watch It Master It 4. [-/0.25 Points] DETAILS SCALCET9 8.3.023.MI. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER The masses m, are located at the points P. Find the moments M, and My and the center of mass of the system. m 1 = 2, m2 = 3, m3 = 5; P1 (2, - 5 ), P2 ( - 3, 3 ), P3 ( 3, 5) My ( x, V ) = Need Help? Read it Watch it Master It