Section 6.3 Estimating Distance Traveled Distance Traveled Suppose a man is driving a car and we know his velocity. Suppose further that we have a graph of his velocity curve from time t = a to time t = b. Question: How do we determine the distance the man traveled during that total time span? Answer: The distance traveled is the area under the velocity curve. Estimating Area/Distance with Rectangles Suppose v(t), the velocity, is a function defined on [a, b]. We can divide the interval [a, b] into n subintervals of equal width Δx =(b - a)/n. We let x 0 = a, x 1 ,x 2 ,...,x n = b be the endpoints of these subintervals. The subintervals will look like . We can estimate the area under the velocity curve (distance) by dividing the area into n rectangles where the width of each rectangle is Δx =(b - a)/n and the heights are given by v(x 0 = a),v(x 1 ),v(x 2 ),...,v(x n = b). In other words, the heights are the velocity function evaluated at the endpoints. If we add up the areas of the rectangles, then we will have an estimate for the area under the curve ( the distance). Note: We are finding the distance on the interval [a, b], using n rectangles, each with a width of Δx, which is given by Δx = (b - a) n Left-Hand Sum: The left-hand sum, L n , is what we calculate when we use only the left endpoints to estimate the area, and is given by L n = Δx (v(x 0 = a)+ v(x 1 )+ v(x 2 )+ ... + v(x n-1 )) Right-Hand Sum: The right-hand sum, R n , is what we calculate when we use only the right endpoints to estimate the area, and is given by R n = Δx (v(x 1 )+ v(x 2 )+ ... + v(x n = b)) I " - E¥xD . Ix . . x xn - - b - - - - - - - - - - w - w u w - - w w w -
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Section 6.3 Estimating Distance Traveled
Distance Traveled Suppose a man is driving a car and we know his velocity. Suppose further that
we have a graph of his velocity curve from time t = a to time t = b.
Question: How do we determine the distance the man traveled during that total time span?
Answer: The distance traveled is the area under the velocity curve.
Estimating Area/Distance with Rectangles
Suppose v(t), the velocity, is a function defined on [a, b]. We can divide the interval [a, b] into n
subintervals of equal width �x = (b� a)/n. We let x0 = a, x1, x2, . . . , xn = b be the endpoints of these
subintervals.
The subintervals will look like .
We can estimate the area under the velocity curve (distance) by dividing the area into n rectangles
where the width of each rectangle is �x = (b� a)/n and the heights are given by
v(x0 = a), v(x1), v(x2), . . . , v(xn = b). In other words, the heights are the velocity function evaluated
at the endpoints. If we add up the areas of the rectangles, then we will have an estimate for the area
under the curve ( the distance).
Note: We are finding the distance on the interval [a, b], using n rectangles, each with a width of �x,
which is given by
�x =(b� a)
n
Left-Hand Sum: The left-hand sum, Ln, is what we calculate when we use only the left endpoints to