Lesson 2 of 7 · 7 min
Horizontal strips
NCERT §8.2
Along one side of the garden the path is straight and the hedge curves away from it. Measured from the path, the hedge is easier to describe as a distance x for each position y.
The lesson in notes
In short
Some regions are easier to slice sideways. If the curve is given as x = g(y), with g ≥ 0, the region between it, the y-axis and the horizontal lines y = c and y = d is cut into horizontal strips of length x and thickness dy.
Adding those strips gives A = ∫꜀ᵈ x dy = ∫꜀ᵈ g(y) dy. The roles of x and y are simply swapped.
Worked example: the region between x = y², the y-axis and the line y = 2 has area ∫₀² y² dy = 8/3.
The same region with vertical strips needs the curve written as y = √x. The strip height is 2 − √x, from x = 0 to 4: ∫₀⁴ (2 − √x) dx = 8 − 16/3 = 8/3, the same answer with more work.
Worked example: the region between x = eʸ, the y-axis and the lines y = 0 and y = 1 has area ∫₀¹ eʸ dy = e − 1.
Choose the strip that needs a single formula for its length across the whole region. If the top edge of a vertical strip changes formula part way along, horizontal strips are often simpler, and the other way round.