Difficulty: Intermediate | Prerequisites: Indefinite integration (power rule, exponential rule, u-substitution)
Definite integrals give integration a concrete numerical answer rather than a family of functions. They are the tool for computing accumulated quantities: area under a curve, area between two curves, total displacement, total revenue. The Riemann sum is the conceptual foundation, approximating the integral by adding up rectangles.
A definite integral evaluates an antiderivative at two endpoints and subtracts. Riemann sums approximate this by slicing the interval into rectangles. The area between two curves is the integral of (top function minus bottom function). Displacement equals the definite integral of velocity.
Definite integral
The integral of f(x) from a to b, written with limits of integration. It produces a single number equal to the net signed area between the curve and the x-axis over that interval. Think of it as "how much has accumulated between a and b."
Riemann sum
An approximation of a definite integral formed by dividing the interval into n subintervals, evaluating the function at a chosen point in each (left endpoint, right endpoint, or midpoint), multiplying by the width, and adding up. In simple terms, it is the total area of a set of rectangles used to estimate the area under a curve.
Fundamental Theorem of Calculus
The theorem connecting differentiation and integration. It states that if F is an antiderivative of f, then the integral from a to b of f(x) dx equals F(b) - F(a). In simple terms, to evaluate a definite integral, find the antiderivative and plug in the endpoints.
Area between curves
The area enclosed between two functions f(x) and g(x) from x = a to x = b, computed as the integral of (top function minus bottom function). In simple terms, you subtract the lower curve from the upper curve and integrate.
Displacement
The net change in position of a moving object, found by integrating the velocity function over the time interval. In simple terms, integrating velocity gives you how far (and in which direction) something has moved.
Divide [a, b] into n equal subintervals. The width of each is delta x = (b - a) / n.
Use the left endpoint of each subinterval to evaluate f(x).
Sum = delta x · [f(x_0) + f(x_1) + ... + f(x_{n-1})].
Example: f(x) = 3x^2 + 2x + 3 on [0, 5] with n = 5. delta x = 1. Left endpoints: 0, 1, 2, 3, 4. f(0) = 3, f(1) = 8, f(2) = 19, f(3) = 36, f(4) = 59. Sum = 1 · (3 + 8 + 19 + 36 + 59) = 125.
The width delta x for an integral from 2 to 10 with n = 4 subintervals is (10 - 2)/4 = 2.
Find the antiderivative F(x), then compute F(b) - F(a). No +C is needed because it cancels.
Example: the integral from 1 to b of (2x - 4) dx. Antiderivative: x^2 - 4x. Evaluate: (b^2 - 4b) - (1 - 4) = b^2 - 4b + 3. Setting this equal to 9 gives b^2 - 4b - 6 = 0, solved by the quadratic formula.
Displacement = the integral from t = a to t = b of v(t) dt.
This is a direct application of the Fundamental Theorem of Calculus.
Example: v(t) = t^2 + 6. Displacement from t = 2 to t = 6 is the integral from 2 to 6 of (t^2 + 6) dt. This can equivalently be written as the integral from 0 to 6 minus the integral from 0 to 2 (the additive property of integrals).
Find the intersection points by setting f(x) = g(x) and solving.
Determine which function is on top over the interval.
Integrate (top - bottom) from the left intersection to the right intersection.
Example: f(x) = x^2 - x - 5 and g(x) = x + 3. Setting equal: x^2 - 2x - 8 = 0, so (x - 4)(x + 2) = 0, giving x = -2 and x = 4. Testing x = 0: g(0) = 3 > f(0) = -5, so g is on top. Area = integral from -2 to 4 of (-x^2 + 2x + 8) dx = 36 square units.
Total revenue over a range = the definite integral of MR(x) from the first unit to the last.
Example: MR(x) = 500 e^(-0.01x). Total revenue from boxes 101 to 200 = the integral from 101 to 200 of 500 e^(-0.01x) dx = -50000 e^(-0.01x) evaluated from 101 to 200 = 50000(e^(-1.01) - e^(-2)) which is roughly $11,445.
\Delta x = \frac{b - a}{n}\text{Left Riemann Sum} = \Delta x \sum_{i=0}^{n-1} f(x_i)\int_a^b f(x)\,dx = F(b) - F(a)\text{Area between curves} = \int_a^b [f(x) - g(x)]\,dx \quad \text{where } f(x) \geq g(x)\text{Displacement} = \int_a^b v(t)\,dtStudents often include +C in definite integrals. The constant cancels when you subtract F(a) from F(b), so it is not written.
Students confuse left and right Riemann sums. A left sum uses the left endpoint of each subinterval; a right sum uses the right endpoint. The exam will specify which.
Students sometimes subtract the curves in the wrong order for area between curves, producing a negative answer. Always subtract the lower function from the upper function.
Students confuse displacement with distance. Displacement is the net change (can be negative); distance is the total path length (always positive). The exam typically asks for displacement.
⚠️ Riemann sum computation (calculating delta x, listing endpoints, evaluating and summing) is a standard short-answer question worth 4 marks.
⚠️ Area between curves is a long-answer staple. You must show intersection points, identify which function is on top, set up the integral, and evaluate. Partial credit is given for each step.
⚠️ The connection between integrals from a to b and the additive property (splitting at an intermediate point) is tested in multiple choice. Know that the integral from 2 to 6 equals the integral from 0 to 6 minus the integral from 0 to 2.
⚠️ Definite integrals of exponential functions (marginal revenue problems) appear regularly. Be comfortable with the antiderivative of e^(ax).
Fill in the blank: delta x for the integral from 2 to 10 with n = 4 subintervals is ______. (2)
True or False: You need to include +C when evaluating a definite integral. (False.)
Fill in the blank: Displacement from t = a to t = b equals the ______ of velocity over that interval. (definite integral)
True or False: When f(x) >= g(x), the area between the curves is the integral of [g(x) - f(x)] dx. (False. It is [f(x) - g(x)] dx.)
Fill in the blank: To find intersection points of two curves, set them ______ and solve. (equal to each other)
Q: If a left Riemann sum uses n = 4 subintervals to approximate the integral from 2 to 10 of f(x) dx, what is delta x?
A: delta x = (10 - 2) / 4 = 2.
Q: Which expression represents the displacement of a particle with velocity v(t) = t^2 + 6 from t = 2 to t = 6?
A: Both the integral from 2 to 6 of (t^2 + 6) dt and the equivalent split form (integral from 0 to 6 minus integral from 0 to 2) are correct.
Q: When f(x) >= g(x), which integral gives the area between the curves from a to b?
A: The integral from a to b of [f(x) - g(x)] dx.
Q: Find the area enclosed by f(x) = x^2 - x - 5 and g(x) = x + 3.
A: Intersection at x = -2 and x = 4. Integral from -2 to 4 of (-x^2 + 2x + 8) dx = 36 square units.
Q: Find the total revenue from selling boxes 101 through 200 if MR(x) = 500 e^(-0.01x).
A: Integral from 101 to 200 of 500 e^(-0.01x) dx = 50000(e^(-1.01) - e^(-2)), roughly $11,445.
Definite integrals rely on the antiderivative skills from the Integration Techniques notes. Area between curves and displacement problems combine those skills with the algebra of setting up the problem (finding intersections, identifying bounds). Total revenue from marginal revenue ties integration to the business-application side of the course, alongside elasticity and optimisation.
definite integral, Riemann sum, left Riemann sum, right Riemann sum, Fundamental Theorem of Calculus, FTC, area between curves, area under curve, displacement, velocity integral, marginal revenue integral, delta x, subinterval width, MAT 210, brief calculus, ASU, net area, total revenue