y = x = (b) Using the formula for a circle centered at the origin with...

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y = x = (b) Using the formula for a circle centered at the origin with radius 3, which is x+ y = 9, identify the lower half y a function of x and separately x as a function of y. A(y) = W = 0 X 3 x-axis (x, y) (c) By revolving the sample rectangular slice (with thickness dy) about the y-axis, we get a cylinder with height dy and volume V = A(x) dy, where A(x) = x is the area of the circular top of the cylinder with radius x. Since we are integrating along the y-axis, use part (b) to write the area as a function of y. dy 0 (x, y) (d) Using the formula, set up but do NOT evaluate the integral needed to determine the work required to pump all the water in the filled hot tub out the top of a standpipe rising vertically two meters out of the top of the hot tub. Use the fluid mass density of water 1000 Kg/m and magnitude of gravity g = 9.8 m/s. (1) Mass from linear density. Given a rod whose density function p(x) (in mass per unit length) measures the density at the value x over a region x = a to x = b, mass is defined by mass = L To visualize this with example sets of units: p(x) dx. (1) mass= dx; a Kg m Kg/m (2) mass = - 1 p(x) dr dx a Oz. oz./in in Consider a thin rod oriented on the positive x-axis over the interval [0, 2/2] with linear density p(x) = = 1 4+x Determine the mass of the rod. (2) Mass from radial density. Given a radially (varying) density function p(x) of a disk of radius r, mass is defined by (the shell method) mass = =2xxp(x) dr. To visualize this with a set of example units: 2 = 2x = p(x) dr mass= Kg a m m Kg/m Consider a plate oriented on the x-axis, resting flat so the center of the plate is at the point (0,0) with radial density p(x) = 3x+1. Calculate the mass of a plate with radius 3 meters. (5) Pumping - Integrating along the y-axis. Consider a full watering trough with an isosceles trapazoidal face. The trough is 4 meters tall, 4 meters wide at the top, 2 meters wide at the bottom, and 5 meters long. Set up AND evaluate the integral needed to deter- mine the work required to pump all the water out over the top edge of the trough. Use the water mass density 1000 Kg/m and magnitude of gravity g = 9.8 m/s. Hint: Imagine the bottom of the face of the trough sitting on the x-axis symmetric about (and parallel to) the y-axis. Just like in problem (4), draw a sample slice, label points, identify the needed formula for the boundary and rewrite it as x = ... y-axis (0, 0) x-axis (3) Work. For a variable force, work is defined as the integral of force over the given distance: -L W = F(x) dx. Written again to visualize common units units (in meters): W Joules = 1 FC a F(x) dx Newtons meters According to Hooke's law, the force required to compress or stretch a spring from its equilibrium to position x is given by F(x)=kx; With units: F(x) = Newtons Newtons/meters meters for the spring constant k. The value of k is unique to each spring, and it is always positive. (a) A spring can be stretched 0.2 meters from its equilibrium position with force of 40 N. Determine the spring constant. (b) The same spring from part (a) has a natural (total) length of 7 meters. The spring is then stretched to the total length of 10 meters. Determine how much work was required to stretch the spring from its natural length of 7 m to a total length of 10 m. - (4) Pumping Integrating along the y-axis. Consider a hot tub in the shape of a hemi- spherical bowl with radius 3 meters buried so the open top is at ground level. Let's determine how much work would be required to pump all the water out the top of a standpipe rising vertically two meters out of the top of the hot tub, using the formula: W = L . gp D(y) A(y) dy, where g is the magnitude of gravity (either 9.8 m/s or 32 ft/s), p is the fluid mass density, D(y) is the pumping distance, and A(y) is the cross-section radius area, and a, b are the lower, upper bounds of the fluid, resp. (a) Consider the image of the hot tub from the side positioned symmetric about the y-axis so the top is on the x-axis (at ground level). Also drawn is a sample rectangular slice (with thickness dy) with labels and the corresponding generic point (x, y). y-axis Height to which we pump, 2 meters above the ground (x-axis). 2 m -3 D(y) dy 0 X 3 x-axis (x, y) (i) With the help of a neighbor, write a brief description of y in this context. (ii) With the help of a neighbor, write a brief description of x in this context. (iii) Determine the length of the arrow labeled D(y), which is the distance the water needs to be pumped from the sample slice up and out the top of a standpipe. y = x = (b) Using the formula for a circle centered at the origin with radius 3, which is x+ y = 9, identify the lower half y a function of x and separately x as a function of y. A(y) = W = 0 X 3 x-axis (x, y) (c) By revolving the sample rectangular slice (with thickness dy) about the y-axis, we get a cylinder with height dy and volume V = A(x) dy, where A(x) = x is the area of the circular top of the cylinder with radius x. Since we are integrating along the y-axis, use part (b) to write the area as a function of y. dy 0 (x, y) (d) Using the formula, set up but do NOT evaluate the integral needed to determine the work required to pump all the water in the filled hot tub out the top of a standpipe rising vertically two meters out of the top of the hot tub. Use the fluid mass density of water 1000 Kg/m and magnitude of gravity g = 9.8 m/s. (1) Mass from linear density. Given a rod whose density function p(x) (in mass per unit length) measures the density at the value x over a region x = a to x = b, mass is defined by mass = L To visualize this with example sets of units: p(x) dx. (1) mass= dx; a Kg m Kg/m (2) mass = - 1 p(x) dr dx a Oz. oz./in in Consider a thin rod oriented on the positive x-axis over the interval [0, 2/2] with linear density p(x) = = 1 4+x Determine the mass of the rod. (2) Mass from radial density. Given a radially (varying) density function p(x) of a disk of radius r, mass is defined by (the shell method) mass = =2xxp(x) dr. To visualize this with a set of example units: 2 = 2x = p(x) dr mass= Kg a m m Kg/m Consider a plate oriented on the x-axis, resting flat so the center of the plate is at the point (0,0) with radial density p(x) = 3x+1. Calculate the mass of a plate with radius 3 meters. (5) Pumping - Integrating along the y-axis. Consider a full watering trough with an isosceles trapazoidal face. The trough is 4 meters tall, 4 meters wide at the top, 2 meters wide at the bottom, and 5 meters long. Set up AND evaluate the integral needed to deter- mine the work required to pump all the water out over the top edge of the trough. Use the water mass density 1000 Kg/m and magnitude of gravity g = 9.8 m/s. Hint: Imagine the bottom of the face of the trough sitting on the x-axis symmetric about (and parallel to) the y-axis. Just like in problem (4), draw a sample slice, label points, identify the needed formula for the boundary and rewrite it as x = ... y-axis (0, 0) x-axis (3) Work. For a variable force, work is defined as the integral of force over the given distance: -L W = F(x) dx. Written again to visualize common units units (in meters): W Joules = 1 FC a F(x) dx Newtons meters According to Hooke's law, the force required to compress or stretch a spring from its equilibrium to position x is given by F(x)=kx; With units: F(x) = Newtons Newtons/meters meters for the spring constant k. The value of k is unique to each spring, and it is always positive. (a) A spring can be stretched 0.2 meters from its equilibrium position with force of 40 N. Determine the spring constant. (b) The same spring from part (a) has a natural (total) length of 7 meters. The spring is then stretched to the total length of 10 meters. Determine how much work was required to stretch the spring from its natural length of 7 m to a total length of 10 m. - (4) Pumping Integrating along the y-axis. Consider a hot tub in the shape of a hemi- spherical bowl with radius 3 meters buried so the open top is at ground level. Let's determine how much work would be required to pump all the water out the top of a standpipe rising vertically two meters out of the top of the hot tub, using the formula: W = L . gp D(y) A(y) dy, where g is the magnitude of gravity (either 9.8 m/s or 32 ft/s), p is the fluid mass density, D(y) is the pumping distance, and A(y) is the cross-section radius area, and a, b are the lower, upper bounds of the fluid, resp. (a) Consider the image of the hot tub from the side positioned symmetric about the y-axis so the top is on the x-axis (at ground level). Also drawn is a sample rectangular slice (with thickness dy) with labels and the corresponding generic point (x, y). y-axis Height to which we pump, 2 meters above the ground (x-axis). 2 m -3 D(y) dy 0 X 3 x-axis (x, y) (i) With the help of a neighbor, write a brief description of y in this context. (ii) With the help of a neighbor, write a brief description of x in this context. (iii) Determine the length of the arrow labeled D(y), which is the distance the water needs to be pumped from the sample slice up and out the top of a standpipe.