The values of m so that the line y=mx and the curve y=xx2+1 enclose a region and find the area of the region
The curve and the line enclose a region when 0<m<1
Area = ln1m-1+m
Area is the integral of the difference of two functions.
To find the intersection points of the given curves, equate both to each other.
Multiply by x2+1
Subtract by x.
x=0 or mx2+m-1
Solve for x by quadratic formula.
For 0<m<1, the intersection of two functions will be at x=0, and at x=±k where k=1m-1.
Since the region is symmetric, there will be two equal areas.
By using ∫-aaf(x)dx=2∫0af(x)dx
By solving the integral,
By substituting limits,
By using k=1m-1
Therefore, the values of m so that the line y=mx and the curve y=xx2+1 enclose a region bounded 0<m<1and area is ln1m-1+m
The enclosed region is 0<m<1