MTH6131 Chap.14 The Definite Integral and the Fundamental Theorem
The Definite Integral and the Fundamental Theorem
From sums to a number
The course outline in the opening lecture lists integrability and the Fundamental Theorem of Calculus among its topics. The definite integral of f from a to b is the limit of sums of rectangle areas under the curve as the rectangles narrow, and a function continuous on the interval is integrable there.
Left sums undershoot and right sums overshoot for a rising curve, and both close in on the same limit. Area below the axis counts as negative, so the definite integral is a signed area.
The Fundamental Theorem
If F is any antiderivative of a continuous f, then the integral from a to b equals F(b) − F(a). This single statement replaces every limit of sums with one subtraction.
The definite-integral exercises open with exactly this instruction, and their solutions write the antiderivative in a bracket with the two limits before showing the subtraction, which is the layout where sign errors become visible.
The theorem also runs the other way: the derivative of the integral from a to x of f is f(x), which the differential equation exercises use to verify solutions defined by integrals.
Substitution, parts and length
In a definite integral a substitution changes the limits along with the variable, so nothing has to be substituted back.
The exercises also use integration by parts in differential form, a half-angle substitution for rational functions of sine and cosine, and two applications: the area under a curve given parametrically, and the length of a curve from the integral of the square root of 1 plus the derivative squared.
A check worth making every time
A positive integrand cannot give a negative integral, and an integrand that is unbounded inside the interval may not have an integral at all.
Checking the sign and the rough size of every answer against the integrand catches the commonest errors, including some that appear in printed worked solutions.
What this chapter covers
- 01
Riemann sums and the limit that defines the integral
- 02
Signed area and the properties of the integral
- 03
The Fundamental Theorem and the bracket layout
- 04
Differentiating an integral with a variable limit
- 05
Substitution with changed limits
- 06
Integration by parts in definite form
- 07
Arc length and parametric area
- 08
Checking the sign and size of an answer
Worked example · free
Two definite integrals with a substitution
- 2Evaluate the first with the Fundamental Theorem.
- 2Substitute u = x² in the second and change the limits.
- 1Check both answers for sign and size.
Key terms
- Definite Integral
- The limit of sums of signed rectangle areas under a curve between two limits, evaluated in practice with the Fundamental Theorem.
- Riemann Sum
- A sum of rectangle areas approximating a definite integral, with heights taken at chosen points in each strip.
- Fundamental Theorem
- The result that the definite integral of a continuous function equals the change in any antiderivative between the limits.
- Arc Length
- The length of a curve between two points, found by integrating the square root of 1 plus the square of the derivative.
- Integration by Parts
- The rule that the integral of u dv equals uv minus the integral of v du, used for products of different kinds of function.
The Definite Integral and the Fundamental Theorem FAQ
Why is there no plus C in a definite integral?
Because the constant cancels. The Fundamental Theorem subtracts the antiderivative at the lower limit from its value at the upper limit, and any constant added to the antiderivative appears in both terms and disappears. Any antiderivative gives the same answer.
Do I need to substitute back after a substitution in a definite integral?
No, provided you change the limits when you change the variable. Once the limits are values of the new variable, the integral is complete in that variable and can be evaluated directly. Substituting back and keeping the old limits also works, but mixing the two is a frequent error.
How can a definite integral be negative?
When the curve lies below the axis over some or all of the interval, because the integral counts that area as negative. It is also negative when the limits are reversed. A positive function integrated from a smaller to a larger limit, though, can never give a negative answer, which makes that a useful check.
Exam move
For every definite integral you evaluate, write one sentence checking the sign and rough size of the answer against the integrand. Then practise substitution with changed limits until changing them in the same line as the variable is automatic.
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