A bicyclist is biking at a rate of (12-ln(x+1)) m.p.h. where x is in hours. After biking for eight hours (he likes biking) how far has he gone?
The bicyclist's velocity is shown by the graph above. The area bounded by the curve, Y=0, x=0 and x=8 gives the total distance traveled by the bicyclist after 8 hours. In order to find the area, an integral needs to be used. The integral that would model this problem is given by:
∫(12-ln(x+1) )dx evaluated from x=0 to x=8
The answer to this integral is: 84.225 miles. The bicyclist has traveled about 84.225 miles after eight hours. This total is equivalent to the area that is swept over by the red line.