Let *R* be the shaded region graphed below.

Use integration with horizontal strips to find the area of *R*.

Hint

What are the limits of integration?

Answer

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* = 2*x*. If *y* = 2*x*, 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