Chapters 10, 12, and 15 all use the same picture to answer a probability question. You shade a region under a curve, then read the shaded area as the answer. That works because area under a density curve is a proportion. That fact is worth re-anchoring now, before later chapters rely on it without comment. This idea was first taught in Calculus II. This course builds only on Probability directly, though. So the review below draws on this section's own check bank, not on Calculus II's questions.
Picture the whole curve's area as a single pie, worth 100% of the outcomes. Shading part of the curve just cuts a slice from that pie. The slice's size is the proportion of outcomes it stands for. Nothing tricky about that.
Consider a continuous variable with density f(x) on an interval. The proportion of outcomes falling between c and d equals the area under the curve there. That area is \displaystyle\int_c^d f(x)\,dx. The total area under the whole curve is exactly 1. This matches the "everything sums to 1" rule from the last section. Here, it's written as an integral instead of a finite sum.
Shading a probability as area
Drag the interval endpoints and watch the shaded area — and the probability readout — change together.
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Worked example
Proportion above a threshold
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1
A quantity is spread evenly over [0, 2], so its density is the constant f(x) = 0.5. What proportion of outcomes exceed 1.5?
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2
The region from 1.5 to 2 is a rectangle: width 2 - 1.5 = 0.5, height 0.5.
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3\boxed{0.5 \times 0.5 = 0.25 - \text{25\% of outcomes exceed 1.5}}
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Worked example
Total area confirms a valid density
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1
Confirm that f(x) = 0.5 on [0,2] is a valid density.
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2
Total area is base times height for this constant curve: (2-0)(0.5) = 1.
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3\boxed{\text{Valid} - \text{the whole area is exactly 1, as any probability density needs}}
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Where this goes next
Confidence intervals and p-values, chapters 09-13
A p-value is literally a shaded tail area under a curve. A confidence interval's coverage is a shaded middle area. Neither of those later pictures needs new mathematics beyond what's here. They just need this exact "area equals proportion" fact. Here, it applies to a specific curve instead of a flat one.
Check your understanding (counts toward your progress)
A density curve has f(2) = 1.4. What does this number represent?
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A single point under a continuous density has probability 0 — f(2) is a density, not a probability.
A density value can legitimately exceed 1; only a probability itself is capped at 1.
Why
A density's height is not a probability — only the area under it over an interval is. Densities routinely exceed 1 as long as the total area still equals 1.
Explain, in the language of this section, why the total area under any valid density curve must equal exactly 1.
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First mistake
Correct answer
Also accepted:
Why
Why
The total area represents the proportion of ALL outcomes, and every outcome is accounted for somewhere under the curve — so that total proportion must be 100%, i.e. an area of exactly 1, the continuous version of a discrete distribution's probabilities summing to 1.
A density is uniform on [0, 5]. What proportion of outcomes exceed 4?
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Why
Height is 1/5 = 0.2; width from 4 to 5 is 1; area = 0.2.
Correct answer
Also accepted:
Why