![]() What would you guess is the exact area, based on where the estimates are headed? Notice that the area is negative, since the graph dips below the x axis. Are the left and right estimates the same? Why? Increase the number of intervals, up to 1000. Select the fourth example, showing a parabola that dips below the x axis.Can you use formulas from geometry to calculate an area for this semicircle? Increase the number of intervals and watch what happens. ![]() Why are the left and right estimates the same? (Hint: use the choice box to show only the left or only the right rectangles and see how they relate). Select the third example, showing a semi-circle (click Equalize Axes if it looks squished).From geometry, you know that the area of a triangle is 1/2 base times height, so the exact area under this curve is 2. Increase the intervals and watch what happens to the left and right estimates. Select the second example from the drop down menu.While we don't know the exact value for the area under this curve over the interval from 1 to 2, we know it is between the left and right estimates, so it must be about 0.69, to two decimal places. Increase the intervals to 4, 10, 100, then 1000. The applet shows a graph of a portion of a hyperbola defined as f ( x) = 1/ x.Later we will learn how to compute the limits in some cases to find a more exact answer. We can calculate the value of a definite integral using a calculator or software and letting n be some large number, like 1,000. An example of an integral for a function of x is, which means to divide up the interval from 0 to 2 into subintervals, sum up the areas of these rectangles (where the height is just x²), and take the limit of this sum as the number of subintervals goes to infinity. Also note that the variable does not have to be t or time. Note that in the limit as n approaches infinity, the left-hand and right-hand Riemann sums become equal. The dt tells you which variable is being integrated (which will not be of much importance until you get to multivariable calculus). The s-shaped curve is called the integral sign, a and b are the limits of integration, and the function f ( t) is the integrand. This is called the definite integral and is written as: If we take the limit as n approaches infinity and Δ t approached zero, we get the exact value for the area under the curve represented by the function. These sums, which add up the value of some function times a small amount of the independent variable are called Riemann sums. We can then write the left-hand sum and the right-hand sum as: Let t i be the ith endpoint of these subintervals, where t 0 = a, t n = b, and t i = a + iΔ t. Divide this interval into n equal width subintervals, each of which is wide. Let f ( t) be a function that is continuous on the interval a ≤ t ≤ b. On this page we will generalize this and write it more precisely. We saw that as we increased the number of intervals (and decreased the width of the rectangles) the sum of the areas of the rectangles approached the area under the curve. On the preceeding pages we looked at computing the net distance traveled given data about the velocity of a car.
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