MATH1202 Chap.9 Integration and Areas
Integration and Areas
This chapter treats integration as both reverse differentiation and accumulation. Indefinite integrals produce antiderivative families with a constant, while definite integrals produce signed totals over stated bounds. Graph crossings determine when a calculation must be split, and the chapter distinguishes signed accumulation from geometric area below an axis or between curves.
Each antiderivative is checked by differentiation, turning a potentially fragile sign or index calculation into a reversible process. Sketches establish which curve is upper and where a function changes sign before any bounds are substituted. Units are interpreted as products of the vertical and horizontal quantities.
What this chapter covers
- 01
Power-rule antiderivatives
- 02
Constant of integration
- 03
Fundamental Theorem of Calculus
- 04
Bound substitution
- 05
Signed versus geometric area
- 06
Area between curves
Worked example · free
Compare signed integral and total area
- +1Find the axis crossing and an antiderivative.
- +1Evaluate the signed definite integral.
- +1Split the region at the crossing and reverse the negative contribution.
- +1Add geometric magnitudes and state square units.
Key terms
- Antiderivative
- A function whose derivative is the stated integrand.
- Integration Constant
- The arbitrary constant required in an indefinite antiderivative family.
- Definite Integral
- Signed accumulation between lower and upper bounds.
- Geometric Area
- The non-negative size of a region after splitting at sign or order changes.
Integration and Areas FAQ
Why does an indefinite integral need a constant?
Differentiation removes every constant, so infinitely many functions share the same derivative. The integration constant records that family. Omitting it gives only one member rather than the complete antiderivative.
Why can a definite integral be negative?
A definite integral is signed accumulation. Contributions below the horizontal axis are negative, and they can outweigh positive contributions. Geometric area instead uses non-negative magnitudes after splitting the interval.
How do I choose top minus bottom?
Sketch the curves, solve their intersections and test which curve is higher on each interval. If their order changes, split the calculation so the integrand remains upper curve minus lower curve.
What is the fundamental theorem doing in a definite integral?
It connects accumulation with antiderivatives. Find an antiderivative, evaluate it at the upper bound and subtract its value at the lower bound. Keep brackets around each substitution, especially when a bound is negative. Differentiating the antiderivative checks the integration step before numerical evaluation begins.
When must an area calculation be split?
Split whenever the integrand changes sign or the identity of the upper curve changes. Find those crossing inputs algebraically and place them in order within the requested interval. A single signed integral may cancel positive and negative pieces, so separate pieces are necessary when the question asks for total geometric area.
How can I estimate a definite integral before calculating?
Sketch the function over the interval and bound the region with simple rectangles or familiar shapes. The integral should have the predicted sign and a plausible magnitude compared with width times typical height. This estimate detects reversed bounds, missing interval splits and arithmetic answers that are inconsistent with the visible region.
What units should an integral have?
Integration multiplies the integrand's units by the input units. Integrating a rate in dollars per week over weeks gives dollars; integrating a height in metres over horizontal metres gives square metres. State the accumulated quantity in context, because the numerical value alone does not reveal whether it is signed change or geometric area.
How do reversed bounds affect a definite integral?
Interchanging upper and lower bounds changes the sign of a definite integral because the direction of accumulation reverses. It does not create a new geometric region. If the task asks for area, first order the boundary inputs and then decide how signs or upper-lower relationships require splitting. A sketch and an estimate should agree with the final sign and magnitude.
Exam move
Differentiate every antiderivative as an immediate check, including the coefficient and changed index. Keep a separate line for the integration constant in indefinite work. For definite integrals, write brackets around upper and lower substitutions and estimate the sign and magnitude before using a calculator.
Before an area calculation, sketch the region, solve crossings and decide whether the question asks for signed accumulation or geometric area. Shade each piece and label which rule is upper on that interval. Practise one problem where a curve crosses the axis and one where two curves exchange order, because these require different-looking integrands but the same splitting logic.
Keep exact values until the final line, attach units that reflect the accumulated quantity, and compare the result with the initial sketch.
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