Let R be the shaded region graphed below.
Use integration with horizontal strips to find the area of R.
What are the limits of integration?
We can slice R into horizontal strips of thickness Δ y:
The right endpoint of the strip at position y is on the line y = 2x. If y = 2x, then , so the coordinates of this point are .
The length of the strip is approximately .
This means the area of the strip at position y is approximately
Summing the areas and taking the limit as the number of strips goes to ∞, we find that the area of R is