Area Examples

What is the area of the shaded region?

If we slice the region into n strips of width Δ x, the area of the strip at coordinate x is approximately

x²Δ x.

If we add up the areas of all n strips between x = 0 and x = 1, we get the approximation

for the area of the shaded region. Taking the limit as n approaches ∞, the exact area of the shaded region is

What is the area of the shaded region?

If we slice the region into strips of width Δ x, the strip at position x looks approximately like a rectangle with height (x – x²) and width dx. The area of that strip is approximately

(x – x²) Δ x.

If we add the areas of all the strips and take the limit as the number of strips approaches ∞, we get

as the area of the shaded region.

Instead of making vertical fish sticks, we could make longer fish sticks using horizontally sliced intervals. In terms of areas under curves, we'd just have to integrate with respect to y instead of x.

Find the area of the shaded region, using horizontal strips.

We slice the region into horizontal strips of width Δ y. When we look at the horizontal strip at height y, the strip lies between two points, each with y as its second coordinate:

The left endpoint of this strip is on the graph of the line y = x, so its full coordinates are (x,y) = (y,y). The right endpoint of the strip is on the graph of the curve y = x². If y = x² then , so the full coordinates of the right endpoint are .

The length of the strip is approximately :

The area of the strip is approximately

If we sum up the approximate areas of the strips and take the limit as the number of strips goes to ∞, we find that the area of the entire region is

Thankfully, this also equals