Areas of regions

Find the area of the following regions. In each case, graph the relevant functions and show the region in question.

The region bounded by the curves y = 1/x, y = x/(3x +4), and the line x =10.

Problem 40E

Find the area of the following regions. In each case, graph the relevant functions and show the region in question.

The region bounded by the curves y = 1/x, y = x/(3x +4), and the line x =10.Solution:-Step 1The graph of

The graph of y = isThe graph of y=10

The graph of the region bounded by these curves isStep 2We must find the point of intersection of different curves with each other and X-axis to find the region bounded by them.

Point of intersection for curves are calculated as follows,1. and y=10 For 2. y = and Equating both the functions, we getEither 3. Also, y = passes through the origin.Step 3From

The bounded region is between . Here y= 10 is above is below ,so area is calculated as follows = = = = = = Putting the limits, we get = = 0.998

This is the area bounded by y=10 and y=From x= to , the area bounded is between and y=. Here is above and y=is below , so area is calculated as follows,

= = = = = = = 3.688-1.3+0.5839 = 2.972

Total region bounded by these curves is = 0.998+2.972= 3.96

The region bounded by the curves y = 1/x, y = x/(3x +4), and the line x =10 is 3.96.