Calculus I · Unit 3A · lesson
The Definite Integral as Signed Area
Learn The Definite Integral as Signed Area through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Section overview
Riemann sums and the definite integralWhat this section is building
Learn The Definite Integral as Signed Area through clear explanation, worked examples, visual reasoning, checks, and connected integral-calculus practice.
Partition, sample, multiply height by width, add, and then refine; the sum approaches a signed accumulated value.
Choose left, right, or midpoint samples from the prompt, predict bias from monotonicity, and distinguish net signed area from geometric area.
Using the wrong endpoints, losing the common width, or adding magnitudes when the integral requires signed contributions.
Learning objectives
Interpret positive and negative contributions and distinguish net signed area from total geometric area.
The Definite Integral as Signed Area
The x-axis determines the sign of contribution
For a graph above the -axis, a definite integral agrees with ordinary geometric area. Below the axis, the function values are negative, so the corresponding contributions are negative. The integral therefore measures signed or net area, not total geometric area. Regions on opposite sides of the axis can cancel even though both occupy physical space on the page.
When a problem asks for total area, first find every crossing of the axis and split the interval. Either reverse the sign on below-axis pieces or integrate the absolute value. Keeping these two questions separate -- net accumulation versus total geometric area -- prevents a very common error in which a perfectly evaluated integral answers the wrong question.
When , the definite integral equals ordinary area under the curve. When , rectangle heights are negative, so those contributions subtract. Thus
Use geometry rather than antiderivatives
Suppose forms a triangle of area above the axis on and a semicircle of area below the axis on . Then
The total geometric area is .
u3a-signed-area-01A graph encloses area 5 above the axis and area 7 below the axis. What is the definite integral?
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Subtract the below-axis area from the above-axis area.
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Sketch a function with but .
Explain why represents total geometric area between the graph and the axis.
Use symmetry to evaluate .
Read this graph as text
Positive and negative signed regions. A curve crosses the x-axis; areas above count positively and areas below count negatively. Display a curve crossing the axis with positive regions above and negative region below. Do not call the integral total geometric area.
Every relationship in positive and negative signed regions uses written labels together with distinct line styles, markers, or fill patterns; color is never the only carrier of meaning.
Why it matters: Display a curve crossing the axis with positive regions above and negative region below.
Positive and negative signed regions. Display a curve crossing the axis with positive regions above and negative region below.
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