DPST1012 Chap.10 Integration and the Fundamental Theorem
Integration and the Fundamental Theorem
Area as a limit
Topic 7, integral calculus, runs in Week 9 and is examined in the final exam. The syllabus lists the fundamental theorem of calculus, computations with definite and indefinite integrals, and the geometry of the definite integral with related applications. The starting question is the area under a curve between two vertical lines.
Thin rectangles give estimates that improve as they narrow, and the definite integral is the limit of those sums.
The fundamental theorem
The theorem connects the two halves of calculus: if F is an antiderivative of f, then the definite integral of f from a to b is F(b) − F(a). Integration therefore undoes differentiation.
An indefinite integral is a family of antiderivatives and carries a constant of integration; a definite integral is a number, because the constant cancels.
The theorem also says that differentiating an accumulated integral with a variable upper limit returns the integrand.
Signed area and applications
A definite integral adds signed area, so regions below the x-axis count as negative and can cancel regions above it. When a question asks for an area, the roots must be found first and each piece integrated separately.
The area between two curves is the integral of the upper curve minus the lower one between their intersection points, and integrating a rate such as a velocity gives the total change in the quantity. A table of standard integrals is provided in the final exam, so choosing and applying the right entry is the skill being tested.
What this chapter covers
- 01
Rectangle estimates of area
- 02
The definite integral
- 03
Antiderivatives and the constant of integration
- 04
The fundamental theorem of calculus
- 05
Standard integrals
- 06
Signed area and area below the axis
- 07
Area between two curves
- 08
Rates and accumulated change
Worked example · free
Evaluate an area that crosses the x-axis
- 1x² − x − 2 = (x − 2)(x + 1), so the curve crosses the axis at x = 2 inside the interval.
- 1An antiderivative is F(x) = x³/3 − x²/2 − 2x, with F(0) = 0, F(2) = 8/3 − 2 − 4 = −10/3 and F(3) = 9 − 4.5 − 6 = −3/2.
- 2From 0 to 2 the integral is −10/3, an area of 10/3 below the axis; from 2 to 3 it is −3/2 − (−10/3) = 11/6.
- 1Total area = 10/3 + 11/6 = 31/6 ≈ 5.17 square units.
Key terms
- Antiderivative
- A function F whose derivative is f, so that integrating f gives F plus a constant.
- Definite Integral
- The signed area between a curve and the x-axis over an interval, computed as F(b) − F(a).
- Indefinite Integral
- The family of all antiderivatives of a function, written with a constant of integration C.
- Fundamental Theorem
- The result that a definite integral equals the change in any antiderivative between the limits.
- Signed Area
- Area counted as positive above the x-axis and negative below it, which is what a definite integral adds up.
Integration and the Fundamental Theorem FAQ
Why does a definite integral not need + C?
The constant appears in both F(b) and F(a) and cancels when one is subtracted from the other. Any antiderivative gives the same definite integral, so the simplest one, with C equal to zero, is used.
Can an integral be negative when an area cannot?
Yes. An integral adds signed area, so a region below the x-axis contributes a negative amount. An area is always positive, so take absolute values of each piece after splitting the interval at the roots.
Will I need to memorise integration formulas?
The assessment map says a table of standard integrals is provided in the final exam. What you need is fluency in choosing the right entry and applying it, which comes from practising with the table beside you.
How do I find the area between two curves?
Find where the curves meet, decide which is on top between those points, then integrate the top curve minus the bottom one. A quick sketch settles which curve is upper and prevents a negative answer.
How does a rectangle estimate relate to the exact integral?
Rectangles that use right endpoints overestimate the area under a rising curve and left endpoints underestimate it. As the rectangles narrow, both estimates approach the same value, and that shared limit is the definite integral given by the fundamental theorem.
Exam move
For every integral you compute, differentiate your answer to check it, and for every area question sketch the curve and mark the roots before writing a single integral sign. Practise with a printed table of standard integrals, as in the exam room.
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