Turn your head 90°: thin horizontal strips: For the area between a curve and the y-axis, just turn the picture 90°: the strips become horizontal.
Area to the x-axis (you already know)
- A stack of thin vertical strips.
- Each has height y and width dx.
- Area = the integral of y dx.
Area to the y-axis
- A stack of thin horizontal strips.
- Each has width x (its distance across to the y-axis) and tiny height dy.
- Add them up: area = the integral of x dy.
Two things change vs. the x-axis version: 1. The variable. Everything inside the integral must be in terms of y, so rearrange y = f(x) to get x = g(y).
2. The limits. They are y-values (the bottom and top of the region), not x-values.
IB-style question — area to the y-axis
The curve y = x² (for x ≥ 0) and the y-axis enclose a region between y = 0 and y = 4.
Find the exact area of this region.
Step by step
- We integrate x dy, so first make x the subject. Since x ≥ 0 we take the positive root.
- Write the area as an integral in y, with the y-limits 0 and 4.
- Integrate y1/2: add 1 to the power and divide by the new power.
- Substitute the limits. 43/2 = (√4)³ = 2³ = 8.
Final answer
A = 16/3 (≈ 5.33) square units.
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The only new skill: make x the subject: The integral needs x in terms of y. So the real work is rearranging.
Make x the subject
- y = x³ → x = y1/3
- y = ln x → x = eʸ (undo ln with the exponential)
- y = eˣ → x = ln y (undo the exponential with ln)
Watch the limits too: If you're given x-limits, convert them to the matching y-values before integrating.
IB-style question — a logarithm curve
The region is bounded by the curve y = ln x, the y-axis, and the lines y = 0 and y = 1.
Find the exact area between the curve and the y-axis.
Step by step
- Rearrange to x = g(y). Undo the natural log with the exponential.
- The y-limits are already given: 0 and 1. Set up the integral of x dy.
- The integral of eʸ is just eʸ.
- Substitute: e¹ − e⁰ = e − 1.
Final answer
A = e − 1 (≈ 1.72) square units.
IB-style question — converting x-limits to y-limits
A region is bounded by y = x² (x ≥ 0), the y-axis, and runs from x = 0 to x = 3.
Using horizontal strips, find the area between the curve and the y-axis.
Step by step
- Rearrange to x = g(y).
- Convert the x-limits to y-limits using y = x². When x = 0, y = 0; when x = 3, y = 9.
- Integrate x dy between the y-limits.
- 93/2 = (√9)³ = 3³ = 27.
Final answer
A = 18 square units.