To determine
To find:
The area of the shaded region.
Answer
36
Explanation
1) Concept:
The area A of a region bounded by the curves y=f(x), y=g(x) and the lines x=a and x=b, where f and g are continuous and f(x)≥ g(x) for all x in a, b is
A= ∫abfx-gxdx
2) Given:
3) Calculation:
From the given graph,
we can see that y=2x is the top (upper) boundary curve and y=x2-4x is the bottom (lower) boundary curve.
Let, fx=2x and gx=x2-4x
0, 0 and (6, 12) are the pointsof intersection of the given curves.
The shaded region lies between x=0 and x=6
Formula-
Area A= ∫abfx-gxdx
So, the area of the shaded region is
A= ∫062x-(x2-4x) dx
=∫066x-x2dx
=6x22-x33 06
Simplifying,
=3x2-x33 06
By taking the upper and the lower limits of integration,
=362-(6)33-0
=3·36-2163
=108-2163
=108-72
=36
Therefore, the area of the shaded region is 36.
Conclusion:
Therefore, the area of the shaded region is 36.