It is often said that we lose ourselves in history, or in the world, or even in our own thoughts, but what kind of loss are we referring to? The lost where you are so enmeshed in the story, so overwhelmed that you can't find the path up, the path down, or even any idea of ​​where to turn next? Or is it the lost where you are so emerged into the storyline, into the character's personal life and feelings, that you hope you never have to resurface again, that you can just continue living in the world of the books of stories. This is how I am in mathematics: I want to be sucked into the world, the language of mathematics, the thousands of theorems and formulas and laws, and I want to never come out again. Say no to plagiarism. Get a tailor-made essay on “Why Violent Video Games Should Not Be Banned”? Get an original essay This article will combine several calculus ideas that all arise from two main topics: the integral and the derivative. The derivative is the instantaneous rate of change of a function at a specific point. Another way to think about the derivative is how the graph changes as x increases or decreases. The integral, or primitive as it is commonly called, is the opposite because it can be seen as canceling differentiation. A scientist measures the depth of the Doe River at Picnic Point. The river is 24 feet wide at this location. Measurements are taken along a straight line perpendicular to the edge of the river. The data is presented in the table below. The water velocity at Picnic Point, in feet per minute, is modeled as v(t) = 16 + 2sin( ) for 0 ≤ t ≤ 120 minutes. Distance from river edge (feet) 0 8 14 22 24 Water depth (feet) 0 7 8 2 0a) Use a trapezoidal sum with the four subintervals indicated by the data in the table to approximate the section area cross section of the river at Picnic Point, in square feet. Show the calculations that lead to your answer. The first step is to know what the trapezoidal ruler is and what it does. The trapezoid rule divides the area under the curve of a definite integral into trapezoids of varying widths and heights depending on the data provided. The areas of these trapezoids are then added to find an approximation of the total area under the curve. The known and accepted formula for the area of ​​the trapezoid is〖Area〗_T=1/2 (h_1+h_2) where h1 and h2 represent the heights of the trapezoids, and w represents the width of the trapezoids. The area of ​​the first trapezoid, with heights 0 and 7 and a width of 8, is 28 ft2, the area of ​​the second trapezoid, with heights 7 and 8 and a width of 6, is 45 ft2, the area of ​​the third trapezoid with heights 8 and 2 and a width of 8, is 40 ft2, and the area of ​​the fourth and final trapezoid, with heights 2 and 0 and a width of 2, is 2 ft2. The area under the curve, using the trapezoidal rule, can be found using the equation〖Area〗_Total=A_1+A_2+A_3+A_4so the total cross-sectional area of ​​the river is 115 ft2.b) The volumetric flow rate at a location along the river is the product of the cross-sectional area and the water velocity at that location. Use your approximation from part (a) to estimate the average value of the volumetric flow rate at Picnic Point, in cubic feet per minute, from t = 0 to t = 120 minutes. The average value of volumetric flow rate can be calculated by the equation 1/ (ba) ∫_a^b▒〖c*f(x)□(24&dx)〗where a is the starting point, b is the end point, c is a constant and f(x) is the equation. This is the mean value theorem and can be used because the function is continuous on the closed interval of [a,b] and differentiable on the intervalopen (a,b). In the context of this problem, the starting point is 0, the ending point is 120, the constant is the cross-sectional area of ​​the river found in the letter a, 115 ft2, and the equation is the equation given for the velocity water from the river. , v(t) = 16 + 2sin( ). The calculation of the average value is then 1/(120-0)∫_0^120▒〖115*v(t)□(24&dt)〗and is evaluated as equal to 1807.17 and the units are ft3/min based on the units of the constant (ft2) multiplied by the units of the equation (ft/ min).c) The scientist proposes the function f, given by f(x) = as a model for the depth of water, in feet , at Picnic Point x feet from the river's edge. Find the cross-sectional area of ​​the river at Picnic Point based on this model. The cross-sectional area can be calculated by integrating the given equation from the starting value to the final value. The general equation of integration is ∫_a^b▒〖f(x)□(24&dx)〗where a is the starting point, b is the end point, and f(x) is the given equation. In the context of the problem, the starting point would be 0 and the ending point would be 24 because that is the total width of the river, and the equation would be the equation given above. Using the equation and data provided, the cross-sectional area can be 122.23 ft2.d) Recall that the volumetric flow rate is the product of the cross-sectional area and the velocity of the water at a place. To prevent flooding, water must be diverted if the average value of volumetric flow at Picnic Point exceeds 2,100 cubic feet per minute during a 20-minute period. Using your answer from part (c), find the average value of the volumetric flow rate during the time interval 40 ≤ t ≤ 60 minutes. Does the value indicate that the water should be diverted?The average value of the volumetric flow rate can be calculated using the same formula as above1/(ba) ∫_a^b▒〖c*f(x)□ (24&dx)〗but this time using the values ​​of 40 minutes for a, 60 minutes for b, 122.23 ft2 as constant and the same speed equation, v(t) = 16+2sin(√(x+10) ), the average value of the volumetric flow rate is 2181.89 cfm. It would therefore be necessary to divert the flow of this part of the river.2. There are 700 people waiting in line for a popular ride at an amusement park when the ride starts operating in the morning. Once operational, the ride accepts passengers until the park closes 8 hours later. As long as there is a queue, people get on the ride at a rate of 800 people per hour. The graph below shows the speed, r(t), at which people arrive at the ride throughout the day. Time t is measured in hours from the time the ride starts operating. How many people arrive at the ride between t = 0 and t = 3? Show the calculations. The number of runners arriving at the race between t=0 and t=3 can be calculated using the trapezoidal rule and the graph below. This is possible because the area under the curve over the given interval corresponds to the number of runners arriving during that period. Two trapezoids can be made on the given interval, one from t=0 to t=2 and the second on the interval t=2 to t=3. For the first trapezoid, the heights are 1000 and 1200, and the width is 2 so the area is 2200, and for the second trapezoid the heights are 1200 and 800, with a width of 1 so the area is 1000. Therefore, the result obtained the total area under the curve and the number of passengers arriving at the route between t=0 and t=3 was 3200 passengers. Does the number of people queuing to board the ride increase or decrease between t = 2 and t = 3? ? Justify. The number of people waiting to board the ride increases between t=2 and t=3 because the rate at which passengers board the ride is 800 passengers per hour.