Math 231E Fall 2013 Lecture 6A Area Definition of Integral Riemann Sums Question What is an integral and why do we care Answer 1 A tool to compute a complicated expression made up of smaller pieces Example 25 Some things that we use integrals to compute 1 Let us say that we are given a function y f x and we want to know the area under the curve 4 3 2 1 0 1 2 3 4 f HxL 2 Let us say that we are given some crazy curve 20 15 10 5 2 How long is the curve 4 6 8 10 12 x 3 Given the dimensions of a tub filled with water how much water does it hold How much work does it take to move all of that water out 4 How much energy does it take to leave a planet s gravity well What s the escape velocity of Earth The main idea behind all integration comes in three steps 1 Break an object into n pieces 2 Approximate those pieces by something we can compute buy exploiting some kind of smallness 3 Add those pieces together Take limit as n 0 1 Area under a curve Let us say that we want to find the area under a curve 1 2 1 0 8 0 6 0 4 0 2 0 0 1 2 3 4 Specifically we are given some function f x and we want the area under the curve from x 0 to x 4 Step 1 Break it into n pieces We will first break this area up into n vertical strips If there are n of them then they have to each be x 4 n wide so we must consider the domains k 1 x k x k 1 2 3 n Notice that the first subdomain is k 0 which is 0 x 0 4 n and the last subdomain is 4 4 4 4 4 n 1 x n x n 1 n n n n For example if we choose n 20 then one of these domains is 8 5 9 5 and we need to compute the area under the curve on this subdomain 1 2 1 0 8 0 6 0 4 0 2 0 0 1 2 3 4 We can blow this picture up and concentrate on the region of interest 1 2 1 0 8 0 6 0 4 0 2 0 1 5 1 6 1 7 1 8 1 9 Step 2 Approximate this area exploiting a smallness Notice that if we can compute the area under the curve for the subdomain 8 5 9 5 1 6 1 8 and all the other areas then we can get the total area by adding all of these up However this problem isn t any simpler than the original problem We make it simpler by approximating Let us replace the area on the subdomain we considered above with a rectangle The question arises of course which rectangle We have several choices 1 2 1 2 1 1 0 8 0 8 0 6 0 6 0 4 0 4 0 2 0 2 0 1 5 1 6 1 7 1 8 1 9 0 1 5 1 6 1 7 1 8 1 9 In the left figure we decided to use the height of the function on the left hand end of the domain to define the rectangle i e the height of the rectange is defined to be f 1 6 This will not surprisingly be called the left hand rule In the right figure we decided to use the height of the function on the right hand end of the domain to define the rectangle i e the height of the rectange is defined to be f 1 8 This will not surprisingly be called the right hand rule There are other possibilities We could choose the midpoint of the interval and define the height to be f 1 7 This is the midpoint rule We could average the two values at the left and right hand endpoints i e choose the height to be 1 f 1 6 f 1 8 2 This is called the averaging rule or the trapezoid rule the motivation for the first name should be obvious the second is also once one sees the correct picture More on this below We can choose the largest value the function ever attains in the interval i e we choose the height to be maxx 1 6 1 8 f x This is called the upper rule We can choose the smallest value the function ever attains in the interval i e we choose the height to be minx 1 6 1 8 f x This is called the lower rule All of these rules will give slightly different answers But they give an answer that we can compute More generally when computing the area associated to the domain k x k 1 x we choose x k k x k 1 x by some rule and then approximate the area by Area width height x f x k Step 3 Add together take limit The area of the kth rectange is f x k x Thus our approximation to the total area is n X k 1 f x k x f x 1 x f x 2 x f x n x Of course this is an incorrect number We have replaced each real small area with a rectangular area But notice that as we take more and more subdomains the approximation error becomes smaller and smaller So there is a hope that if we take the limit as the number of subdomains goes to infinity Thus we define Z 4 n X f x dx lim f x k x n 0 k 1 if it exists There is another subtle problem to worry about Notice that before we make some set of choices for the x k to get the rectangle heights Clearly this will affect the computation but if we are doing the right thing then maybe it shouldn t matter So let us make a definition Definition 5 A function is called integrable on the domain a b if the procedure described above break into pieces approximate rectangles by some choice then take limit always gives the same answer Theorem 11 Every continuous function or even piecewise continuous function is integrable This theorem is very useful since it tells us that we can use whatever rectangle rule we find convenient for a given problem and this won t mess up the answer Let us work out an example We want to compute Z 1 x2 dx 0 We first do the case where n 4 to warm up then compute the general n formula and then the limit If n 4 x 1 4 and our four subdomains are 0 1 4 1 4 1 2 1 2 3 4 3 4 1 The left hand rule says that the four rectangle heights we will use are f 0 0 f 1 4 1 …
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