What is an Integral? That’s because we’ve been talking about area, which is always positive. The definite integral of a positive function f(x) over an interval [a, b] is the area between f, the x-axis, x = a and x = b.; The definite integral of a positive function f(x) from a to b is the area under the curve between a and b.; If f(t) represents a positive rate (in y-units per t-units), then the definite integral of f from a to b is the total y-units that accumulate between t = a and t = b The integral is also called as anti-derivative as it is the reverse process of differentiation. Work can also be calculated from integrating a force function, or when counteracting the force of gravity, as in a pumping problem. The change in population =  \int_{1980}^{1990} f(t)dt = -\text{area between} f \text{and axis} $. The top here is a curve, so we can’t get an exact answer. If the velocity is positive, positive distance accumulates. https://www.whitman.edu/mathematics/calculus_online/chapter09.html My estimate is that about 312 calls were made between 9 pm and 11 pm. I’ll choose to use 4 rectangles, and I’ll choose left-endpoints:  \int_{9}^{11} r(t)dt \cong 100(.5) + 150(.5) + 180(.5) + 195(.5) = 312.5$. If the function is positive, the signed area is positive, as before (and we can call it area.). Integration is basically used to find the areas of the two-dimensional region and computing volumes of three-dimensional objects. Know how to calculate average values Apply integration to the solution of engineering problems Negative rates indicate that the amount is decreasing. The total number of calls will be  \int_{9}^{11} r(t)dt $. For example, if f(t) is the velocity of a car in the positive direction along a straight line at time t (miles/hour) , then negative values of f indicate that the car is traveling in the negative direction, backwards. The units are calls per hour × hours = calls. Write a definite integral to represent the total change in the duck population from 1980 to 1990, and estimate the population in 1990. Be able to split the limits in order to correctly find the area between a function and the x axis. Definite integrals can be used to determine the mass of an object if its density function is known. A bug starts at the location x = 12 on the x–axis at 1 pm walks along the axis with the velocity v(x) shown in figure 6. 1. Approximately how many calls were made between 9 pm and 11 pm? If the velocity is negative, distance in the negative direction accumulates. Is this an under-estimate or an over-estimate? How wide are the rectangles? We know that the accumulated calls will be the area under this rate graph over that two-hour period, the definite integral of this rate from t = 9 to t = 11. We’ll expand our idea of a definite integral now to include integrands that might not always be positive. The table shows rates of population growth for Berrytown for several years. The area between the velocity curve and the x-axis, between 2:30 and 3, shows the total distance traveled by the bug in the negative direction, back toward home; the bug traveled 2.5 feet in the negative direction. 231 CHAPTER 6 Applications of the Deﬁnite Integral in Geometry, Science, and Engineering EXERCISE SET 6.1 1. (Opens a modal) Exploring accumulation of change. Use Figure 7 to calculate  \int_{0}^{2}f(x)dx$,  \int_{2}^{4}f(x)dx $,  \int_{4}^{5}f(x)dx$, and  \int_{0}^{5}f(x)dx $. The “heights” of the rectangles, the values from the function, now might not always be positive. If f(t) represents a positive rate (in y-units per t-units), then the definite integral of f from a to b is the total y-units that accumulate between t = a and t = b. The application of science, technology, and math to design, build, and maintain structures, machines, and processes.  f(x) = \int_{1}^{3} (1 + x)dx$ represents the area between the graph of f(x) = 1+x, the x–axis, and the vertical lines at 1 and 3 (figure 3).

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