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%% Integration of Numeric Data % This example shows how to integrate a set of discrete velocity data % numerically using |trapz| to approximate the total distance traveled. The % |integral| family only accepts function handles as inputs, so those % functions cannot be used with discrete data sets. Use |trapz| when a % functional expression is not available for integration. %% View the Velocity Data % Consider the following velocity data and corresponding time data. % Copyright 2015 The MathWorks, Inc. vel = [0 .45 1.79 4.02 7.15 11.18 16.09 21.90 29.05 29.05 ... 29.05 29.05 29.05 22.42 17.9 17.9 17.9 17.9 14.34 11.01 ... 8.9 6.54 2.03 0.55 0]; time = 0:24; %% % This data represents the velocity of an automobile (in m/s) taken at 1 s % intervals over 24 s. %% % Plot the velocity data points and connect each point with a straight % line. figure plot(time,vel,'-*') grid on title('Automobile Velocity') xlabel('Time (s)') ylabel('Velocity (m/s)') %% % The slope is positive during periods of acceleration, zero during periods % of constant velocity, and negative during periods of deceleration. At % time |t = 0|, the vehicle is at rest with |vel(1) = 0| m/s. The vehicle % accelerates until reaching a maximum velocity at |t = 8| s of |vel(9) = % 29.05| m/s and maintains this velocity for 4 s. It then decelerates to % |vel(14) = 17.9| m/s for 3 s and eventually back down to rest. Since this % velocity curve has multiple discontinuities, a single continuous function % cannot describe it. %% Calculate the Total Distance Traveled % |trapz| performs discrete integration by using the data points to create % trapezoids, so it is well suited to handling data sets with % discontinuities. This method assumes linear behavior between the data % points, and accuracy may be reduced when the behavior between data points % is nonlinear. To illustrate, you can draw trapezoids onto the graph using % the data points as vertices. xverts = [time(1:end-1);time(1:end-1);time(2:end);time(2:end)]; yverts = [zeros(1,24);vel(1:end-1);vel(2:end);zeros(1,24)]; hold on p = patch(xverts,yverts,'b','LineWidth',1.5); %% % |trapz| calculates the area under a set of discrete data by breaking the % region into trapezoids. The function then adds the area of each trapezoid % to compute the total area. %% % Calculate the total distance traveled by the automobile (corresponding to % the shaded area) by integrating the velocity data numerically using % |trapz|. distance = trapz(time,vel) %% % The distance traveled by the automobile in |t = 24| s is about 345.22 m.