Final answer:
To calculate the area enclosed by the curves y=1/x, y=1/x², and x=2, we integrate each function from x=1 to x=2, then subtract the smaller area from the larger area. The total area is given by ln(2) - 1/2.
Step-by-step explanation:
To determine the area of the region enclosed by the curves y=1/x, y=1/x², and x=2, we first need to find the intersections of the curves and set the limits for the area computation. The intersection of y=1/x and y=1/x² occurs at x=1, since substituting x into both equations yields y=1. The region we are interested in is then bounded by these curves between x=1 and x=2. The area under each curve can be found by integrating the function over this interval.
The area under y=1/x from x=1 to x=2 is given by the integral:
A_{1/x} = ∫_{1}^{2} 1/x dx = ln(x) | from 1 to 2 = ln(2) - ln(1) = ln(2)
Similarly, the area under y=1/x² is:
A_{1/x²} = ∫_{1}^{2} 1/x² dx = -1/x | from 1 to 2 = -1/2 - (-1/1) = 1/2
The total area enclosed between the curves is then:
Total Area = A_{1/x} - A_{1/x²} = ln(2) - 1/2