Question

find the area of the region bounded by the lines y=x and y=8x and the curve y=(1/x^2)

Answer 1

The two lines intersect at the origin (0,0).
The line y=8x intersects the curve y=1/x^2 at (1/2,4).
The line y=x intersects the curve at (1,1).
Will split the integrals into two, since the upper limit changes from the steep line to the curve.
\[I1=\int\limits_{0}^{1/2}(8x-x)dx\]
and
\[I2 = \int\limits_{1/2}^{1} (1/x ^{2} - x)dx\]
Evaluating the integrals, we get
I1=7/8 and I2=5/8 for a total area of 3/2.