Calculus 220 section 6 2 Area Under a Curve Riemann Sums notes by Tim Pilachowski Consider the function f x x on the interval 0 10 With the x axis the horizontal line y 0 and the vertical line x 10 f forms a triangle We could find the area of the triangle by counting squares There are 45 full squares and 10 half squares for a total of 50 of them An easier method would be to use knowledge of geometry to calculate the area of that triangle which is also by the way the area under the curve Putting the correct values into 1 the formula A bh we get 2 1 area of triangle area under the curve 10 10 50 2 This scenario is fairly easy because the function f x x is a line and the area under the curve of f forms a well known geometric shape What happens when the curve is not linear but actually curves Since there is no geometric formula for irregularly shaped spaces we ll need a way to approximate the area under the curve Suppose that we form a series of rectangles under f x x on the interval 0 10 and use those to approximate the area under the curve If we draw in 10 rectangles of width 1 and put the midpoint of the top of each rectangle on the line f x x the sum of the areas of the rectangles approximation of the area under the curve 1 0 5 1 1 5 1 2 5 1 3 5 1 4 5 1 5 5 1 6 5 1 7 5 1 8 5 1 9 5 50 In more general terms the interval a b was split up into n subintervals called partitions of b a width x n The height of each rectangle is a y value f x evaluated at the midpoint of the partition Area under the curve f x1 x f x2 x f x3 x K f xn x f x1 f x 2 f x3 K f xn x This formula is called a Riemann sum and provides an approximation to the area under the curve for functions that are non negative and continuous In most of your homework exercises you will be asked to use this midpoint version of a Riemann sum It is also possible to use either the left or right endpoints of the intervals Example A Approximate the area under the curve y 2 x on the interval 2 x 7 using five partitions and left endpoint midpoint and right endpoint sums Example A extended Repeat the approximation process using 10 partitions left endpoint midpoint and right endpoint sums 4 3 3 7 2 which is 3 approximately 20 92244274 Just as increasing the number of partitions brought us closer to the true value for the area under the curve y 2 x it is reasonable to suppose that in general for any function increasing the number of partitions will provide an increasingly better approximation to the area under the curve We will in time look at the limit of a Riemann sum as the number of partitions n approaches The exact value for the area under the curve y 2 x on the interval 2 x 7 is Now comes an important question Why would we be interested in the area under a curve Consider a velocity function v t When v t is constant it is not difficult to see that the formula distance rate of speed time is the area of the rectangle formed on the graph i e distance area under the curve v t v t When v t is changing the area of the rectangles formed by our partitions gives us average rate of speed on the partition time area under the curve distance time We ve already determined that velocity rate of change of distance first derivative of distance This relationship turned backwards is distance antiderivative of velocity Substituting from the observations above we can conclude area under the curve v t antiderivative of v t integral of v t A similar thought process can be applied to other functions such as marginal cost marginal profit and marginal revenue we considered in earlier chapters Marginal cost was defined as the cost to make the next unit or cost per unit Area under the marginal cost curve cost per item number of items cost to produce Area under C x C x antiderivative of C x integral of C x