The piecewiselinear function f is graphed below. If F' = f, and if F(2) = 3, then find each of the values in the table. Enter a numeral in each cell. A rod is 5 meters long and its density at a distance x meters from its left end, in kilograms per meter, is given by the function [900 = 480236 Find the mass of the rod. Round your answer to at least 2 decimal places. Do not enter any units in your answer. Note that the units are already built into the answer! Mass of rod z kg (Enter a decimal number rounded to at least 2 decimal places) Consider the region shown below, which is bounded by the graphs of y = 1n(x), the horizontal axis, and the line x = 3. The region is rotated about the line x = 1 to form a solid of revolution. 35:3 1 l | | l l l l l l l l l l | l I - .> 2 4 B l | | l l l l l l l | l y=|n(x) { l i) Find a definite integral that represents the volume of the solid obtained by revolving the region about the line x = 1. 01\"\") 7r(4)2 7r(1n(y) + 1)2 dy b) f1 7r(1n(x)+1)2 dx o) f13 1n(x) dx d) [13 2m + 1) 1n(x) dx 0111(3) 7r(ey + 1)2 dy II 8) e) i) Find a definite integral that represents the volume of the solid obtained by revolving the region about the line x = - 1. a) Som n(4) 2 - n(In(y ) + 1) dy b) S' x ( In(x ) + 1) dx c) " In(x) dx d) f 2 x(x + 1) In(x) dx e) fan(3) m(ey + 1)2 dy Answer (Select one of the choices from the pull-down menu) ii) Based on part i), what type of volume element is used to set up the integral? Answer: iii) Use a numerical integration routine to calculate the volume of the solid of revolution. Round your answer to at least 2 decimal places. Calculators: TI-84 Desmos Simpson's Rule Approximator Volume ~ (Enter a decimal number rounded to at least 2 decimal places)Suppose that the function f is continuous, decreasing and concave down on some interval [0, 19]. Let LEFT(n), RIGHT(n), M|D(n) and TRAP(n) denote, respectively, the left, right, midpoint and trapezoidal rule estimates with n subdivisions, of b / f(x) dx. Let EXACT denote the exact value of the integral. Arrange these values in increasing order by dragging each one from the list and dropping them in their correct positions in the chained inequality below. An oil tank in the shape of a right circular cylinder with a height of 6 feet and radius of 2 feet is buried so that its top is 3 feet beneath ground level, as shown in the diagram. The depth of oil in the tank is 5 feet, and the oil weighs 50 pounds per cubic foot. round level 3 ft. 6 ft. i) Let x represent the vertical distance from ground level to the surface of the oil as it is being pumped out. Find an integral that represents the total amount of work done in pumping all of the oil up to ground level. a) A5 200m dx i) Let x represent the vertical distance from ground level to the surface of the oil as it is being pumped out. Find an integral that represents the total amount of work done in pumping all of the oil up to ground level. a) [05 200m: dx b) ff 2007rx dx c) [36 ($00xe dx d) [05 507L152 dx e) f 200::(22 x2) dx Answer: (Select one of the choices from the pull-down menu) ii) Use a numerical integration program to find the total work done to pump all of the oil up to ground level. Round your answer to the nearest whole foot-pound. Calculators: TI-84 Desmos Simpson's Rule approximator Total work done z ft-lb (Enter a numeral) Important: Enter a numeral. Do not enter any units in your answer. Note that the units are already built into the answer! Consider the second-order differential equation y" = ky, where k # 0 is a constant. Select all function(s) in the list below that are solutions to this differential equation. There may be more than one correct answer. In the options below, A, B and C are nonzero constants. Select one or more: Oa. y= 0 Ob. y = Ae-Vkx O c. y = A sin (Vkx ) + Bcos(Vkx) O d. y = Levkx + C O e. y = Aev kx Of. y = - sin(Vkx) +c