MTH1020 · Analysis of Change
Integration: The Definite Integral & the FTC
Week 10 of Monash MTH1020 Analysis of Change introduces integration. It builds the definite integral as a limit of Riemann sums and interprets it as signed area, states its properties, and then delivers the Fundamental Theorem of Calculus — both the evaluation form ∫ₐᵇ f = F(b) − F(a) and the derivative form (d/dx)∫ₐˣ f = f(x) — which ties differentiation and integration together. It closes with the indefinite integral and the standard antiderivatives. These ideas are central to the final exam and are the platform for the Week-11 techniques.
What this chapter covers
- 01The definite integral as a limit of Riemann sums: ∫ₐᵇ f dx = lim(n→∞) Σ f(xᵢ*)Δx, Δx = (b−a)/n
- 02LEFT(n) and RIGHT(n) endpoint sums; the limit is independent of the sample-point choice
- 03The integral as signed area: regions above the axis count positively, below negatively
- 04Properties: ∫ₐᵃ f = 0, ∫ₐᵇ f = −∫ᵇₐ f, ∫ₐᵇ f = ∫ₐᶜ f + ∫ᶜᵇ f, linearity
- 05FTC evaluation form: if F′ = f then ∫ₐᵇ f(x) dx = F(b) − F(a) = [F(x)]ₐᵇ
- 06FTC derivative form: (d/dx) ∫ₐˣ f(t) dt = f(x)
- 07The indefinite integral ∫ f(x) dx = F(x) + C (a family of functions), distinct from the definite integral (a number)
- 08Standard antiderivatives: ∫xⁿ dx = x^(n+1)/(n+1) + C (n ≠ −1), ∫(1/x) dx = ln|x| + C, ∫eˣ dx = eˣ + C, ∫cos x dx = sin x + C
Evaluating a definite integral by the FTC, plus the derivative form
- +1(a) Find an antiderivative. Using ∫xⁿ dx = x^(n+1)/(n+1) term by term, an antiderivative of 3x² − 2x is F(x) = x³ − x² (since d/dx[x³ − x²] = 3x² − 2x).
- +1(a) Apply the evaluation form of the FTC. ∫ from 0 to 2 of (3x² − 2x) dx = [x³ − x²] from 0 to 2 = F(2) − F(0).
- +1(a) Substitute the limits. F(2) = 2³ − 2² = 8 − 4 = 4 and F(0) = 0 − 0 = 0, so the integral equals 4 − 0 = 4.
- +1(b) Apply the derivative form of the FTC. Since (d/dx) ∫ from a to x of f(t) dt = f(x), differentiating with respect to the upper limit x gives (d/dx) ∫ from 1 to x of cos(t²) dt = cos(x²). (The lower limit 1 is constant and contributes nothing.)
Key terms
- Riemann sum
- An approximation Σ f(xᵢ*)Δx to the area under a curve using rectangles. The definite integral is its limit as the number of subintervals n → ∞.
- Definite integral
- ∫ₐᵇ f(x) dx, a number equal to the signed area between the graph and the x-axis on [a, b]. Areas above the axis are positive, below negative.
- FTC (evaluation form)
- If f is continuous and F′ = f, then ∫ₐᵇ f(x) dx = F(b) − F(a). The integral of a rate of change equals the total change.
- FTC (derivative form)
- (d/dx) ∫ₐˣ f(t) dt = f(x): differentiating an integral with respect to its upper limit returns the integrand.
- Antiderivative
- A function F with F′ = f. Any two antiderivatives of f differ by a constant, so the indefinite integral is written F(x) + C.
- Indefinite integral
- ∫ f(x) dx = F(x) + C, a family of functions (with the constant of integration C). Distinct from the definite integral, which is a single number.
Integration: The Definite Integral & the FTC FAQ
What's the difference between a definite and an indefinite integral?
A definite integral ∫ₐᵇ f(x) dx has limits and evaluates to a number — the signed area under the graph between a and b. An indefinite integral ∫ f(x) dx has no limits and represents a whole family of antiderivatives, F(x) + C, where the constant of integration C accounts for the fact that any two antiderivatives differ by a constant. The FTC connects them: you evaluate the definite integral by taking an antiderivative and subtracting its values at the two limits.
Why must I include the constant of integration?
Because differentiation destroys constant information — the derivative of x³ − x² and of x³ − x² + 7 are identical — so reversing it can only recover the antiderivative up to an added constant. For an indefinite integral you therefore must write + C to represent every antiderivative. In a definite integral the constant cancels when you subtract F(b) − F(a), so it does not appear in the final number, but omitting it from an indefinite answer is a marks-losing slip.
What does the Fundamental Theorem of Calculus actually say?
It links the two operations of calculus. The evaluation form says that to add up f over [a, b] you only need an antiderivative F and the difference F(b) − F(a) — integration is the reverse of differentiation. The derivative form says that differentiating an accumulated area with respect to its upper limit gives back the original function. Together they mean differentiation and integration undo one another, which is why finding antiderivatives is the key skill for Week 11.
Can Sia help me with integration in MTH1020?
Yes. Sia can check an antiderivative by differentiating it back with you, walk through applying either form of the FTC, and explain the signed-area interpretation step by step. It teaches the method and checks your reasoning on your own practice questions; it does not do graded assessment for you, and Monash academic-integrity rules apply.
Exam move
Make antiderivatives automatic, because the FTC reduces most integration to 'find F, then subtract'. Memorise the standard antiderivatives (including ∫(1/x) dx = ln|x| + C, the one students most often get wrong) and always verify an antiderivative by differentiating it back before substituting limits. Practise both forms of the FTC — evaluation for definite integrals, derivative form for accumulated-area functions — and keep the definite-versus-indefinite distinction sharp (a number versus a family with + C). Understand the signed-area picture so a negative answer makes sense. This chapter underpins the final and is the foundation for the Week-11 techniques, so drill it hard and ask Sia to generate fresh definite integrals and check each line.
Working through Integration: The Definite Integral & the FTC in MTH1020? Sia is AskSia’s AI Mathematics tutor — ask any MTH1020 Integration: The Definite Integral & the FTC question and get a clear, step-by-step explanation grounded in how MTH1020 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
