Area, Volume, and Arc Length Exercises

Answer

The region R is a triangle with height 2 and base 1.

Therefore its area is

Answer

We can 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

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 = 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