Difficulty: Intermediate | Prerequisites: limits, derivatives, basic Riemann sums
The Fundamental Theorem of Calculus (FTC) is the bridge between differentiation and integration, two operations that initially seem unrelated. It sits at the heart of any integral calculus course and is the single most important result you will meet this semester. To follow this material, you should be comfortable with limits, basic derivative rules, and the idea of a Riemann sum. If those feel shaky, revisit them first.
The definite integral is a limit of Riemann sums and gives the net accumulation of a quantity over an interval. The integral function collects that accumulation as you move the upper limit. The Fundamental Theorem says that differentiation and integration undo each other: the derivative of the integral function is the original function, and any antiderivative lets you evaluate a definite integral by plugging in the endpoints.
Definite integral
The value obtained by taking the limit of Riemann sums of f(x) on [a, b]. Formally, ∫ₐᵇ f(x) dx = lim (n→∞) Σᵢ₌₁ⁿ f(xᵢ*) Δx.
Think of it as the total net area between the curve and the x-axis, where area below the axis counts as negative.
Riemann sum
An approximation of the definite integral formed by slicing [a, b] into n subintervals of width Δx = (b − a)/n, evaluating f at a sample point xᵢ* in each subinterval, and summing f(xᵢ*) Δx.
In simple terms, this means chopping the region under a curve into thin rectangles and adding up their areas.
Integral function (accumulation function)
The function I(x) = ∫ₐˣ f(s) ds, which gives the running total of accumulated "stuff" from a to x.
Think of it as a counter that tracks how much has piled up so far. If f is a velocity, I(x) is the distance travelled. If f is a rate of flow, I(x) is the total volume.
Antiderivative
A function F(x) whose derivative equals f(x), i.e. F′(x) = f(x). Antiderivatives are not unique: if F is one, then F + C is another for any constant C.
In simple terms, this means undoing the derivative. For example, both x³ and x³ − 7 are antiderivatives of 3x², because differentiating either one gives you 3x² back.
Dummy variable
The variable of integration inside a definite integral (such as s or t in ∫ₐᵇ f(s) ds). Its name does not matter: ∫ₐᵇ f(x) dx = ∫ₐᵇ f(s) ds = ∫ₐᵇ f(t) dt.
Think of it as a placeholder label. Swapping it out changes nothing about the integral's value.
Partition the interval [a, b] into n equal subintervals, each of width Δx = (b − a) / n.
The left endpoint of the i-th subinterval is xᵢ₋₁ = a + (i − 1)Δx. The right endpoint is xᵢ = a + iΔx.
Choose a sample point xᵢ* in each subinterval and form the Riemann sum: Σᵢ₌₁ⁿ f(xᵢ*) Δx.
The definite integral is defined as the limit of this sum as n → ∞, provided the limit exists.
The variable of integration is a dummy variable. Writing ∫ₐᵇ f(x) dx, ∫ₐᵇ f(s) ds, or ∫ₐᵇ f(t) dt all mean the same thing.
Define I(x) = ∫ₐˣ f(s) ds. This is a function of its upper limit x, not of the dummy variable s.
I(x) gives the running total of whatever f represents, accumulated from a up to x.
If f(x) is the height of a curve, I(x) is the area under that curve between a and x.
If f(x) is velocity, I(x) is the distance travelled between time a and time x.
If f(x) is a rate of production, I(x) is the total output between a and x.
I(a) = 0 always, because you have accumulated nothing when the upper limit equals the lower limit.
F(x) is an antiderivative of f(x) when F′(x) = f(x).
Antiderivatives differ from one another by a constant. If F(x) is one antiderivative, every antiderivative has the form F(x) + C.
Example: x³ and x³ − 7 are both antiderivatives of 3x², because the derivative of each is 3x².
FTC Part 1: The integral function I(x) = ∫ₐˣ f(t) dt is an antiderivative of f. That is, I′(x) = f(x). Differentiation undoes integration.
FTC Part 2: If F is any antiderivative of f on [a, b], then ∫ₐᵇ f(x) dx = F(b) − F(a). This lets you evaluate definite integrals without computing limits of sums.
\Delta x = \frac{b - a}{n}x_{i-1} = a + (i-1)\,\Delta x, \quad x_i = a + i\,\Delta x\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta xI(x) = \int_a^x f(s)\,ds\frac{d}{dx}\int_a^x f(t)\,dt = f(x) \quad \text{(FTC Part 1)}\int_a^b f(x)\,dx = F(b) - F(a) \quad \text{(FTC Part 2)}Engineers use the integral function every time they compute total displacement from a velocity profile, or total energy from a power curve. If you have ever seen a car's trip odometer, that is an accumulation function in action: it integrates speed over time to give distance.
Students often think that the definite integral always equals an area. It does not. When f(x) is negative on part of [a, b], the integral counts that region as negative. The integral gives net signed area, not geometric area.
Students sometimes confuse the dummy variable with the upper limit. In I(x) = ∫ₐˣ f(s) ds, the variable s is a dummy (its name is irrelevant), while x is the real variable that I depends on.
Students frequently forget the "+ C" when writing the general antiderivative. A single function f(x) has infinitely many antiderivatives, each differing by a constant.
Students sometimes believe that the Riemann sum must use right endpoints. Any sample point xᵢ* in [xᵢ₋₁, xᵢ] works: left endpoints, right endpoints, midpoints, or anything else. The limit is the same.
⚠️ Be ready to set up a Riemann sum from scratch: write Δx, the endpoints, the sum, and then take the limit. This is a classic exam problem.
⚠️ Know both directions of the FTC. Part 1 (derivative of an integral) often appears as: "Find d/dx ∫ₐˣ f(t) dt." Part 2 (evaluating a definite integral with an antiderivative) is the workhorse of every integration problem.
⚠️ Expect questions that test whether you understand the integral function conceptually, for example: "If f(t) represents velocity, what does ∫₀⁵ f(t) dt represent?" The answer is total displacement, not total distance (unless f ≥ 0 on the interval).
⚠️ Professors commonly ask students to explain why two antiderivatives of the same function can look different. The answer is always "they differ by a constant."
True or false: ∫ₐᵇ f(x) dx and ∫ₐᵇ f(t) dt always give the same value. (True)
Fill in the blank: Δx = ______ when the interval [a, b] is divided into n equal parts. (Δx = (b − a) / n)
True or false: If F′(x) = f(x), then F(x) is the only antiderivative of f(x). (False, any F(x) + C also works)
Fill in the blank: The integral function I(x) = ∫ₐˣ f(s) ds always satisfies I(a) = ______. (0)
True or false: The definite integral of a function that is negative on [a, b] gives a negative number. (True)
Q: Write the Riemann sum for ∫₁⁴ x² dx using n subintervals and right endpoints.
A: Δx = 3/n. The right endpoint of the i-th subinterval is xᵢ = 1 + 3i/n. The Riemann sum is Σᵢ₌₁ⁿ (1 + 3i/n)² (3/n).
Q: Let I(x) = ∫₂ˣ (t³ − 5t) dt. Find I′(x).
A: By FTC Part 1, I′(x) = x³ − 5x.
Q: Find ∫₀³ (6x² + 1) dx using the Fundamental Theorem.
A: An antiderivative of 6x² + 1 is F(x) = 2x³ + x. Then F(3) − F(0) = (54 + 3) − 0 = 57.
Q: Give two different antiderivatives of f(x) = cos(x) and explain why both are valid.
A: sin(x) and sin(x) + 4. Both are valid because the derivative of each is cos(x). The constant 4 vanishes under differentiation.
Q: If v(t) represents velocity in m/s and ∫₀¹⁰ v(t) dt = 85, what does the number 85 represent?
A: The total displacement of the object over the time interval [0, 10], measured in metres. (If v were negative on part of the interval, some forward distance would be cancelled by backward distance.)
This material connects directly to substitution and integration techniques, because FTC Part 2 is the reason those techniques are useful: they find antiderivatives, and the theorem turns antiderivatives into definite integral values. It also connects to differential equations, where the relationship F′ = f is the starting point for solving equations involving rates of change. Later in the course, the FTC generalises to functions of several variables via line integrals and the theorems of Green, Stokes, and Gauss.
Fundamental theorem of calculus, FTC, FTC Part 1, FTC Part 2, definite integral, Riemann sum, accumulation function, integral function, antiderivative, indefinite integral, net signed area, dummy variable, evaluation theorem, area under a curve, integral calculus, UT Austin integral calculus