Phase 2 - Lesson 2.6

Integration and the Fundamental Theorem of Calculus

The integral as accumulated area, the Fundamental Theorem linking it to antiderivatives, and its use in present value and accumulated cash flows.

⏱ 60 min● Intermediate🔗 Prereqs: 2.5

↖ Phase 2 hub

Builds on: 2.1 supplied limits and 2.2 the derivative; the integral is a different limit that the FTC ties back to the derivative.
Leads to: Integration underpins expectation (Phase 7), the Itô integral (Phase 9), and continuous-time valuation.

Learning Objectives

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Define the definite integral as a limit of Riemann sums and interpret it as signed area.
State both parts of the Fundamental Theorem of Calculus and use them to evaluate integrals.
Apply substitution to compute integrals.
Compute the present value of a continuous cash-flow stream by integration.

Key Vocabulary

Riemann sum: ∑ f(x_i*) Δx, an approximation of area by rectangles over a partition.

Definite integral: ∫_a^b f dx = limit of Riemann sums as the mesh → 0; the signed area under f.

Antiderivative: A function F with F′=f; unique up to an additive constant.

FTC Part 1: d/dx ∫_a^x f(t)dt = f(x): integration and differentiation are inverse operations.

FTC Part 2: ∫_a^b f dx = F(b)−F(a) for any antiderivative F of f.

Present value: Today’s worth of future money, discounted at rate r; a future dollar at time t is worth e^{−rt}.

The integral as accumulated area

Partition \([a,b]\) into slices, approximate the area under \(f\) by rectangles, and refine. The limit - when it exists - is the definite integral, the signed area between the curve and the axis:

\[\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^{n} f(x_i^*)\,\Delta x,\qquad \Delta x=\frac{b-a}{n}\] (2.7)

The Fundamental Theorem

The FTC is the hinge of calculus: it says differentiation and integration undo each other. Part 1: the area-so-far function \(G(x)=\int_a^x f(t)\,dt\) has derivative \(G'(x)=f(x)\). Part 2: to evaluate a definite integral, find any antiderivative \(F\) and subtract:

\[\int_a^b f(x)\,dx=F(b)-F(a),\qquad F'=f\] (2.8)

Worked Example - Evaluate ∫₀² (3x²+1) dx

Antiderivative: \(F(x)=x^3+x\) since \(F'(x)=3x^2+1\).

Apply FTC Part 2: \(F(2)-F(0)=(8+2)-(0+0)=10\).

So the signed area is 10. No rectangles needed once an antiderivative is known.

Substitution reverses the chain rule: for \(\int f(g(x))g'(x)\,dx\), let \(u=g(x)\) so \(du=g'(x)\,dx\), turning it into \(\int f(u)\,du\).

Finance link: present value of a cash-flow stream

Suppose income arrives continuously at rate \(c(t)\) dollars per year, discounted continuously at rate \(r\). A dollar at time \(t\) is worth \(e^{-rt}\) today, so the present value of the whole stream over \([0,T]\) is an integral:

\[\mathrm{PV}=\int_0^T c(t)\,e^{-rt}\,dt\] (2.9)

For a constant stream \(c(t)=c\), this evaluates to \(\mathrm{PV}=\frac{c}{r}\big(1-e^{-rT}\big)\) - the continuous analogue of an annuity, derived in the exercise.

Interactive: Riemann sum in real Python

Common Mistakes to Avoid

Forgetting the +C on an indefinite integral, or worse, carrying it into a definite integral where it cancels.
Not changing the limits (or back-substituting) when using substitution - a and b are x-values, not u-values.
Treating ∫ as always positive; it is SIGNED area and is negative where f<0.
Discounting a continuous stream with a discrete factor instead of e^{−rt}.

Quant Practitioner Tips

FTC Part 2 turns area into an antiderivative evaluation - master a table of antiderivatives.
For substitution, either convert the limits to u or resubstitute x before plugging in; never mix.
Present value is just a discounted integral; the discount factor e^{−rt} is the continuous cousin of 1/(1+r)^t.
Sanity-check any symbolic integral with a quick numerical Riemann sum.

Knowledge Check

Q1 Easy

FTC Part 1 states that d/dx ∫_a^x f(t)dt equals:

F(x)−F(a)

f(x)

f′(x)

Q2 Medium

∫₀¹ (4x³) dx equals:

x⁴

Q3 Medium

The present value of a continuous stream c(t) discounted at rate r over [0,T] is:

∫₀ᵀ c(t)dt

∫₀ᵀ c(t)e^{−rt}dt

∫₀ᵀ c(t)e^{rt}dt

c(T)e^{−rT}

Practical Exercise

A perpetual project pays a constant \(c\) dollars/year continuously, discounted at rate \(r\gt 0\). (a) Compute the present value over \([0,T]\). (b) Take \(T\to\infty\) to get the value of the perpetuity.

▶ Show full solution

(a) With \(c(t)=c\) constant:

\[\mathrm{PV}=\int_0^T c\,e^{-rt}\,dt=c\Big[-\tfrac1r e^{-rt}\Big]_0^T=\frac{c}{r}\big(1-e^{-rT}\big).\]

(b) As \(T\to\infty\), \(e^{-rT}\to0\), so \(\mathrm{PV}\to c/r\). A continuous perpetuity paying \(c\) per year is worth \(c/r\) today - the continuous-time version of the classic perpetuity formula.

After the reveal, answer for yourself: The finite-horizon answer smoothly limits to the perpetuity as T grows. Checking that a general formula degrades to a known special case is a powerful correctness test.

Lesson Summary

The definite integral is a limit of Riemann sums measuring signed area; the Fundamental Theorem of Calculus links it to antiderivatives (Part 1: d/dx of the area function is f; Part 2: evaluate via F(b)−F(a)). Substitution reverses the chain rule. In finance, integrating a cash-flow stream against the discount factor e^{−rt} yields present value, and a constant perpetuity is worth c/r.

Formula Sheet Additions

Fundamental Theorem

\[\int_a^b f=F(b)-F(a),\quad F'=f\]

Antidifferentiation evaluates area.

PV of continuous stream

\[\mathrm{PV}=\int_0^T c(t)e^{-rt}\,dt\]

Discount each instant’s cash flow and accumulate.

Retrieval Practice

Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.

▶ Show retrieval prompts & answers

Q: State both parts of the Fundamental Theorem of Calculus.
A: Part 1: if G(x)=∫_a^x f(t)dt then G′(x)=f(x). Part 2: ∫_a^b f dx=F(b)−F(a) for any antiderivative F of f. Together they make differentiation and integration inverse operations.

Q: How do you compute the present value of a continuous cash-flow stream, and what is a constant perpetuity worth?
A: PV=∫_0^T c(t)e^{−rt}dt; for a constant rate c over an infinite horizon this equals c/r, the continuous perpetuity value.

Completion Checklist

I can explain the core ideas in my own words
I worked the derivations/examples by hand
I completed the interactive workbench(es)
I passed the knowledge check

Confidence / mastery rating

Personal notes

Source References

This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.

Calculus, Vol. I (Tom Apostol, 2nd ed., 1967) foundational - Vol. I, Ch. 1-2, 5 - The integral as a limit of sums and the Fundamental Theorem of Calculus.
Stochastic Calculus for Finance II (Steven Shreve, 2004) foundational - discounting - Continuous discounting and integrals of cash flows as the setting for later risk-neutral valuation.

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