Chapter 14 – 1b Area Between Curves
Chapter 14 – 1b Area Between CurvesExample 1. An amusement company maintains records for each video game it installs in an arcade. Suppose that C(t) and R(t) represent the total accumulated costs and revenues (in thousands of dollars), respectively, t years after a particular game has been installed. IfC't=2 and R't=9e-0.3tfind the area between the graphs of C’ and R’ over the useful life of the game and interpret the results.349123073660059e-0.3t-2dx=-90.3e-0.3t-2t05=-30e-0.3*5-2*5—30e-0.3*0-2*0≈-6.6939-10--30=13.306The total profit generated by the game over the 5-year life of the game is about $13,306.Example 2. A yeast culture is growing at a rate of M't=0.3e0.1t grams per hour. Find the area between the graph of M’(t) and the t-axis over the interval [0,10] and interpret the results.430593500100.3e0.1t=0.30.1e0.1t010=3e1-3e0≈8.15-3.00=5.15The mass of the yeast culture grew by approximately 5.15 grams over the 10 hour period.Example 3. A beginning high school language class was chosen for an experiment on learning. Using a list of 50 words, the experiment involved measuring the rate of vocabulary memorization at different times during a continuous 5-hour study session. It was found that the average rate of learning for the entire class was inversely proportional to the time spent studying and was given approximately by4286885412115V't=15t 1≤t≤5Find the area between the graph of V’(t) and the t-axis over the interval [2,3] and interpret the results.2315tdt=15lnt23=15ln3-15ln2≈6.08On average, the students in the class learned about 6 words during the 3rd hour of the study session.430593514605Example 4. Find the area between the curves y=e0.5x and y=-1x on the interval 1≤x≤2.12e0.5x+1xdx=2e0.5x+lnx12=2e1+ln2-2e0.5+ln1≈6.1297-3.2974≈2.8324305935266700Example 5. Find the area between the curves y=x2-1 and y=1.5x on the interval -3.5≤x≤2.5.-3.5-0.5x2-1-1.5xdx+-0.521.5x-x2+1dx+22.5x2-1-1.5xdx=-3.5-0.5x2-1-1.5xdx--0.52x2-1-1.5xdx+22.5x2-1-1.5xdx=x33-x-0.75x2-3.5-0.5-x33-x-0.75x2-0.52+x33-x-0.75x222.5≈0.271—-19.979—-2.333-0.271+-1.979—-2.333=20.250—-2.604+0.354≈23.21Note: 20.250+-2.604+0.354=18.00. This is the cumulative signed area between the two curves. When the blue curve is above the red curve, the area is positive. When the blue curve is below the red curve, the area is negative.-3.52.5x2-1-1.5xdx=x33-x-0.75x2-3.52.5≈-1.979--19.979 ≈18.00 ................
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