Sometimes tested MM7.4

Properties of Definite Integrals

This topic covers the algebraic rules for manipulating definite integrals. These rules allow you to combine or split up integrals, which often simplifies complex expressions into something much easier to evaluate without a calculator.

Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • If two integrals have identical upper and lower limits, you can combine their functions into a single integral over that same range.
  • If two integrals of the same function have 'contiguous' limits (the upper limit of one is the lower limit of the other), you can join them into a single integral spanning the overall range.
  • Flipping the limits of integration negates the value of the integral. For example, the integral from 5 to 2 is the negative of the integral from 2 to 5.
  • The rule for combining contiguous ranges works regardless of whether the limits are in increasing order. For instance, the integral from 2 to 7 plus the integral from 7 to 4 equals the integral from 2 to 4.

Formulae

∫a^b f(x) dx + ∫a^b g(x) dx = ∫a^b [f(x) + g(x)] dx

Use this to combine or separate different functions being integrated over the exact same interval.

∫a^b f(x) dx + ∫b^c f(x) dx = ∫a^c f(x) dx

Use this to join integrals of the same function over adjacent intervals. Note that b can be numerically between a and c, or outside them.

∫b^a f(x) dx = - ∫a^b f(x) dx

Use this fundamental property to handle integrals where the upper limit is smaller than the lower limit. It is the key to avoiding sign errors.

Definitions

Contiguous Ranges
In the context of integrals, this means the limits of integration link up sequentially, like ∫ from a to b followed by ∫ from b to c.

Worked example

Given that ∫14 f(x) dx = 6 and ∫14 g(x) dx = -2, what is the value of ∫41 [f(x) - 3g(x)] dx?

  1. 1

    First, split the integral using the sum/difference rule:

    ∫41 [f(x) - 3g(x)] dx = ∫41 f(x) dx - 3 ∫41 g(x) dx.

  2. 2

    The limits are from 4 to 1, which is the reverse of the given integrals (1 to 4).

    We must flip the limits and negate the value for each integral.

  3. 3

    So, ∫41 f(x) dx = -∫14 f(x) dx and ∫41 g(x) dx = -∫14 g(x) dx.

  4. 4

    Substitute these into the expression:

    [-∫14 f(x) dx] - 3[-∫14 g(x) dx] = -∫14 f(x) dx + 3∫14 g(x) dx.

  5. 5

    Now substitute the given numerical values:

    -(6) + 3(-2).

  6. 6

    Calculate the final result:

    -6 - 6 = -12

Answer: -12

Common mistakes

  • ×Sign Errors: The most common mistake is forgetting to introduce a negative sign when flipping the limits of integration. For example, wrongly assuming ∫52 f(x) dx is the same as ∫25 f(x) dx.
  • ×Confusion with Contiguous Limits: Hesitating or making errors when the connecting limit is not numerically between the start and end limits (e.g., in ∫15 + ∫53 = ∫13).

No-calculator tips

  • ✓Always check the limits first. Same limits? Combine the functions. Contiguous limits? Combine the range. This is your primary decision.
  • ✓If you see an integral with the upper limit smaller than the lower limit, your first mental step should be to rewrite it as the negative of the integral with the limits swapped. This prevents sign errors propagating through your calculation.

Read this topic in the official UAT-UK ESAT guide →

All Mathematics 2 topics