What Is an Integral?

let's say we want to find the area under this function on the interval from 1 to 9 one way to do this is to divide the region into four segments and to draw a rectangle inside each segment such at the upper right corner of each rectangle touches the function the sum of the area of these rectangles is an estimate of the area under the curve the estimate is obviously poor in this case because the rectangles just don't fit under the curve they're too wide we can do better by doubling the number of rectangles once twice three times four times five times and six times until we have a total of 256 rectangles these narrower rectangles fit under the curve better than the wider ones and so the sum of their areas is a much better estimate of the area under the curve we could do this differently by placing rectangles into the four segments such at the left edge of each rectangle touches the curve we could then double the number of rectangles once twice three times four times five times and six times the result is the same a bunch of rectangles that fit under the curve fairly well and they provide a decent estimate of the area under the curve when the area of all 256 rectangles is summed finally we could fill the four segments with rectangles such that the top center of each rectangle touches the curve doubling the number of rectangles once twice three times four times five times and six times leads to the now-familiar result that the rectangles better fit under the curve allowing the sum of the area of all rectangles to provide a good estimate of the area under the curve it does not matter whether we place the rectangle such that the curve touches them on the upper right corner in the center or on the upper left corner so long as they are narrow enough to fit nicely under the curve they can provide an estimate of the area under the curve what we'd like to do is translate this concept into math looking at the same function again let's imagine we want to find the area under the curve between these two x-coordinates let's further imagine that we will estimate the area using rectangles inside three equally sized regions to translate this into math we need labels for the x coordinates that define the range of each of these regions from left to right let's call them x0 x1 x2 and x3 now let's place a rectangle between x0 and x1 such at the upper right corner of the tangle touches the curve at x1 the area of this rectangle is its height which is the value of the function at x1 times its width which is x1 minus x0 to estimate the area under the curve we need to add to this the area of the rectangle in the next region which is its height the value of the function at x2 times its width x2 minus x1 finally we need to add to this the area of the rectangle in the third region which is the value of the function at X 3 times x3 minus x2 let's call this sum of rectangle areas capital s and give it a subscript of 3 to indicate the number of rectangles used in the sum this can be written more compactly using summation notation with the variable J serving as the subscript and taking values from 1 to 3 the argument to the sum is the height of each rectangle the value of the function at X sub J times the width of each rectangle X sub J for the x-coordinate of the right side of the j3 angle minus the x-coordinate of the left side of the same rectangle which is X sub J minus 1 the width can be written even more compactly as Delta X sub J with the understanding that this stands for X sub J minus X sub J minus 1 3 rectangles do not provide a very good estimate of the area under the curve from X 0 to X 3 and so we might want to include more rectangles to do this take the height of the J 3 angle times its width and sum from J equals 1 to an arbitrary number of rectangles in this is S sub n a sum of the area of n rectangles and is called a Riemann sum the number of rectangles could be 6 12 24 or any other number but S sub n only becomes a good estimate of the area under the curve if we take the limit of this quantity as the number of rectangles the value of n goes to infinity in the limit of including very many rectangles S sub n is the area between the curve and the x-axis from X 0 to X sub N and is written symbolically as the integral from X 0 to X n of the function f of X times an infinitesimal width D X this is the riemann definition of an integral between two fixed limits sometimes called a Riemann integral in the integrand f of X is the height of an infinitesimally narrow rectangle and DX is the infinitesimal width of that tangle as we have learned a Riemann integral can be interpreted geometrically as the area between the curve and the x-axis on an interval but this is only true if the function lies above the x-axis for instance this function lies above the x-axis between x equals 0 and x equals one point 185 the Riemann integral over this interval is positive one point one to one and it can be interpreted as the area under the curve in this interval because the function is never negative between x equals one point one eight five and three the situation is different because the function is negative the Riemann integral over this region is negative two point nine nine seven because the function is always negative clearly a real geometric area can never be negative and so we need to refine our definition of a Riemann integral a better definition of a riemann integral is that it is the net area between the function and the x axis over some interval with the understanding that areas below the x-axis are negative and areas above the x-axis are positive only in the case in which the function is always positive can we interpret the Riemann integral as a true geometric area otherwise the value is a net area so far we have talked about definite integrals ones that are between fixed limits and that are written as the integral from A to B of a function times a differential width where a and B are constants these integrals have been defined in terms of Riemann sums with a geometric interpretation that is helpful but that does not provide a practical way to evaluate integrals this is provided by the fundamental theorem of calculus which we'll get to in just a second if we replace the upper limit B with the variable X we obtain the integral from a to X of the function times a differential width the result is no longer a value instead the result is a function of X that we'll call capital f of X this integral is an indefinite integral because the upper limit is not definite it is important to note that the variable U is just a dummy variable of integration and could be anything you t WX and so on the function f of U is the function f of X with X change to you for convenience and clarity so that X is not confused with the upper limit of integration the fundamental theorem of calculus states that the derivative of the function capital f of X which is the result of an indefinite integral is equal to the function being integrated little f of X this defines differentiation as the inverse of integration and integration as the inverse of differentiation this provides an immediate way to find the indefinite integral of a function determine what function when differentiated gives the function you're trying to integrate for instance what is the integral of cosine X the function that when differentiated gives cosine X is sine X thus sine X must be the integral of cosine X of course adding any constants such as C also gives cosine X when differentiated we sometimes call this the constant of integration and that's what is an integral

Channel: Physical Chemistry

Views: 834,197

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Keywords: area, net area, Riemann sum, Riemann integral, integral, definite integral, indefinite integral, fundamental theorem of calculus, calculus

Id: Stbc1E5t5E4

Length: 7min 22sec (442 seconds)

Published: Tue Aug 18 2015