Hello, I need help with calc 2 Assignment, attached, please show all the work.
1) A vertical plate is partially submerged in water and has the indicated shape. Derive the approximation of the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral . Use the coordinate system that is set - up . Do not solve the integral . ( " = 9800 () / + ! , ! ) _____/10 2) The arc of the parabola . = / ! from ( 1 ,1 ) to ( 2 ,4 ) is rotated about the . - axis. a. Sketch the surface on the graph provided. ____/ 1 b. Find the surface area of the resulting surface of revolution. _____/ 9 3) Set - up the integrals that represent the surface area generated by rotating the given region about the specified axis. Do not solve the integrals. / = 4. " + . # , . = 0 , 0 ≤ . ≤ 1 a) About the / - axis b) About the . - axis _____/ 5 . = / $ ! , . = 0 , 1 ≤ / ≤ 2 c) About the / - axis d) About the . - axis _____/ 5 4) a. Derive the arc length 7 of a function 8 ( / ) from / = 9 to / = : by using Pythagoean Theorem on a differential triangle. That is, use the relationship ( ;, ) ! = ( ;/ ) ! + ( ;. ) ! _____/ 1 ____________________________________________________________________________________ b. Find the length of the curve . = % ! ( < & + < $ & ) from / = 0 to / = ln 2 . ( You must show your steps. ) _____/ 9 5 ) Below is an autonomous 1 st order differential equation that has unique solutions on the entire ?@ - plane. ;?;@ = 5 − ? ! a. Draw a phase portrait. Be sure to sketch a solution ? ( @ ) in each strip of the plane. _____/10 b. Suppose ? = ? ( @ ) is a solution to the differential equation in part a. If the solution satisfies the given initial condition, determine the limit. i) ? ( 0 ) = 2 lim ' → ) ? ( @ ) = ii) ? ( 3 ) = √ 5 lim ' → ) ? ( @ ) = iii) ? ( 0 ) = 6 lim ' → $ ) ? ( @ ) = iv) ? ( − 1 ) = − √ 6 lim ' → $ ) ? ( @ ) = _____/5 6 ) a. Solve the following differential equations equation. Give solution in explicit form. That is, . = 8 ( / ) . / ;.
;/ = 4 . + / * < & , / > 0 _____/ 5 ____________________________________________________________________________ b. Solve the following Initial - Value Problem. You may give solution in implicit form. That is, 8 ( / , . ) = I , I ∈ ℝ / ! ;.
;/ = . − /. , . ( − 1 ) = − 1 _____/ 5 7 ) Find all orthogonal trajectories to the given curve. . = / ! − 5 _____/10 Extra Credit: You must show all work. No partial credit is awarded here. Find the length of the curve / ! / " + . ! / " = 1 . __________________________________________________________________________________________________ _____/1
