Let R be the shaded region graphed below.
Without using calculus, find the area of R.
The region R is a triangle with height 2 and base 1.
Therefore its area is
Use integration with vertical strips to find the area of R.
We slice R into vertical strips of width Δ x:
The strip at position x has approximate height
2 – y = 2 – 2x.
This means the area of that strip is approximately
(2 – 2x) Δ x.
Summing the areas and letting the number of strips go to ∞, we find that the area of R is
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
Make it rain.