Definite Integrals
Definite integrals are a cornerstone of AP Calculus, extending the concept of integration to calculate precise values like the area under a curve, net change, and more. Unlike indefinite integrals, which result in a family of functions (with a "+ C"), a definite integral always yields a single numerical value.
This guide covers key definitions, Riemann sums, properties of definite integrals, the Fundamental Theorem of Calculus Part 2, worked examples, key formulas, memory aids, and a practice quiz.
1Introduction to Definite Integrals
Imagine you want to find the exact area of an irregularly shaped region. For simple shapes like rectangles or triangles, geometry formulas suffice. But what about the area under a curved line? This is where definite integrals come into play.
A definite integral calculates the net signed area between a function's curve and the x-axis over a specified interval. It's a powerful tool that bridges the gap between geometry and calculus, allowing us to quantify accumulation and change in a precise way.
Imagine you're filling a swimming pool with a hose that changes flow rate over time. The definite integral of the flow rate function over a time interval gives you the exact total volume of water added during that period -- even though the rate was constantly changing.
Why It Matters
Physics
Computing displacement from velocity, work from force, and total charge from current over time.
Economics
Finding total revenue from marginal revenue functions, consumer and producer surplus.
Engineering
Calculating areas, volumes, center of mass, and moments of inertia for complex shapes.
Statistics
Computing probabilities from probability density functions over continuous distributions.
2Key Definitions
Definite Integral
Represented by the symbol ∫ from a to b of f(x) dx, it calculates the exact net signed area between the curve of the function f(x) and the x-axis from x = a to x = b.
Net Signed Area: Area above the x-axis is considered positive, and area below the x-axis is considered negative. The definite integral sums these values.
Limits of Integration
The values a and b in ∫ from a to b. The lower limit (a) is the starting x-value, and the upper limit (b) is the ending x-value of the interval.
Integrand
The function f(x) being integrated. It is the expression immediately following the integral symbol.
Differential (dx)
Indicates the variable with respect to which the integration is performed. Conceptually represents an infinitesimally small change in x.
Riemann Sum
An approximation of the area under a curve by dividing it into rectangles and summing their areas. As the number of rectangles approaches infinity, the sum approaches the exact definite integral.
Antiderivative
A function F(x) is an antiderivative of f(x) if F'(x) = f(x). Used to evaluate definite integrals via the Fundamental Theorem of Calculus.
Continuity
A function is continuous on an interval if its graph can be drawn without lifting the pen. Continuity is required for the FTC to apply.
3Riemann Sums: Approximating Area
Before we can find the exact area, it's helpful to understand how we approximate it using rectangles. This concept forms the foundation of the definite integral.
The process involves:
- Dividing the interval [a, b] into n subintervals of equal width, Δx = (b - a) / n.
- Choosing a specific point within each subinterval to determine the height of the rectangle.
- Summing the areas of these rectangles.
General Riemann Sum Formula
A ≈ ∑ from i=1 to n of f(xᵢ*) · Δx
where xᵢ* is the sample point chosen in the i-th subinterval.
Types of Riemann Sums
Left Riemann Sum (LRAM)
The height of each rectangle is determined by the function's value at the left endpoint of each subinterval.
LRAM = Δx [f(x₀) + f(x₁) + ... + f(xₙ₋₁)]
Right Riemann Sum (RRAM)
The height of each rectangle is determined by the function's value at the right endpoint of each subinterval.
RRAM = Δx [f(x₁) + f(x₂) + ... + f(xₙ)]
Midpoint Riemann Sum (MRAM)
The height of each rectangle is determined by the function's value at the midpoint of each subinterval. Often provides a better approximation.
MRAM = Δx [f(x̄₁) + f(x̄₂) + ... + f(x̄ₙ)]
Trapezoidal Rule
Uses trapezoids instead of rectangles by averaging the left and right Riemann sums. Often yields even more accurate results.
T = (Δx/2) [f(x₀) + 2f(x₁) + ... + 2f(xₙ₋₁) + f(xₙ)]
The Definite Integral as a Limit
The definite integral itself is the limit of Riemann sums as the number of subintervals n approaches infinity (and thus Δx approaches 0):
∫ from a to b of f(x) dx = lim(n→∞) ∑ from i=1 to n of f(xᵢ*) · Δx
4Properties of Definite Integrals
Definite integrals obey several useful properties that can simplify calculations and help us manipulate expressions.
1. Order of Integration
Swapping the limits of integration changes the sign of the integral.
∫ from a to b of f(x) dx = -∫ from b to a of f(x) dx
2. Zero Width Interval
If the upper and lower limits are the same, the integral is zero.
∫ from a to a of f(x) dx = 0
3. Constant Multiple Rule
A constant factor can be moved outside the integral.
∫ from a to b of c · f(x) dx = c · ∫ from a to b of f(x) dx
4. Sum/Difference Rule (Linearity)
The integral of a sum or difference is the sum or difference of the integrals.
∫ from a to b of [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
5. Additivity (Splitting the Interval)
If c is a point between a and b, the integral can be split into two parts.
∫ from a to b of f(x) dx = ∫ from a to c of f(x) dx + ∫ from c to b of f(x) dx
6. Comparison Properties
- If f(x) ≥ 0 for a ≤ x ≤ b, then ∫ from a to b of f(x) dx ≥ 0.
- If f(x) ≤ g(x) for a ≤ x ≤ b, then ∫ f(x) dx ≤ ∫ g(x) dx.
- If m ≤ f(x) ≤ M on [a, b], then m(b - a) ≤ ∫ f(x) dx ≤ M(b - a).
Pro Tip
The additivity property is incredibly useful on the AP exam. It lets you split complicated integrals at convenient points, or combine given integral values to find new ones -- even if the splitting point c is not between a and b!
5Evaluating Definite Integrals with Antiderivatives (FTC Part 2)
The most elegant and powerful way to evaluate definite integrals is using the Fundamental Theorem of Calculus, Part 2 (FTC2). This theorem establishes a direct link between definite integrals and antiderivatives.
Theorem Statement (FTC Part 2)
If f is continuous on [a, b] and F is any antiderivative of f (meaning F'(x) = f(x)), then:
∫ from a to b of f(x) dx = F(b) - F(a)
Steps to Evaluate a Definite Integral Using FTC2
- Find the antiderivative: Determine F(x), the indefinite integral of f(x). You do not need to add the "+ C" for definite integrals, as it cancels out.
- Evaluate at the upper limit: Calculate F(b).
- Evaluate at the lower limit: Calculate F(a).
- Subtract: Compute F(b) - F(a).
Notation
We often write F(b) - F(a) as [F(x)] evaluated from a to b, using a vertical bar with the limits. This compact notation is standard in textbooks and on the AP exam.
6Worked Examples
Example 1: Basic Polynomial
Evaluate ∫ from 1 to 3 of (x² + 2x) dx
Step 1: Find the antiderivative
F(x) = x³/3 + x²
Step 2: Evaluate F(b) - F(a)
F(3) = 27/3 + 9 = 9 + 9 = 18
F(1) = 1/3 + 1 = 4/3
Step 3: Subtract
= 18 - 4/3 = 54/3 - 4/3 = 50/3
Example 2: Trigonometric Function
Evaluate ∫ from 0 to π/2 of cos(x) dx
Step 1: Find the antiderivative
F(x) = sin(x)
Step 2: Evaluate F(b) - F(a)
F(π/2) = sin(π/2) = 1
F(0) = sin(0) = 0
Step 3: Subtract
= 1 - 0 = 1
Example 3: Using Properties (Splitting the Integral)
Given ∫ from 0 to 2 of f(x) dx = 5 and ∫ from 2 to 5 of f(x) dx = -3, find ∫ from 0 to 5 of f(x) dx.
Apply the Additivity Property:
∫ from 0 to 5 of f(x) dx = ∫ from 0 to 2 of f(x) dx + ∫ from 2 to 5 of f(x) dx
Substitute the given values:
= 5 + (-3) = 2
Example 4: Exponential Function
Evaluate ∫ from 0 to 1 of e2x dx
Step 1: Find the antiderivative (u-substitution: u = 2x)
F(x) = (1/2)e2x
Step 2: Evaluate F(b) - F(a)
F(1) = (1/2)e²
F(0) = (1/2)e⁰ = 1/2
Step 3: Subtract
= (1/2)e² - 1/2 = (1/2)(e² - 1)
Example 5: Absolute Value (Requires Splitting)
Evaluate ∫ from -1 to 2 of |x| dx
Step 1: Split at x = 0 where the definition changes
|x| = -x when x < 0, and |x| = x when x ≥ 0
∫ from -1 to 2 of |x| dx = ∫ from -1 to 0 of (-x) dx + ∫ from 0 to 2 of x dx
Step 2: Evaluate each integral
∫ from -1 to 0 of (-x) dx = [-x²/2] from -1 to 0 = 0 - (-1/2) = 1/2
∫ from 0 to 2 of x dx = [x²/2] from 0 to 2 = 4/2 - 0 = 2
Step 3: Add the results
= 1/2 + 2 = 5/2
7Key Formulas
Definite Integral (FTC Part 2)
∫ from a to b of f(x) dx = F(b) - F(a), where F'(x) = f(x)
Order of Integration
∫ from a to b of f(x) dx = -∫ from b to a of f(x) dx
Zero Width
∫ from a to a of f(x) dx = 0
Constant Multiple
∫ from a to b of c · f(x) dx = c · ∫ from a to b of f(x) dx
Sum/Difference (Linearity)
∫ from a to b of [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
Additivity
∫ from a to b of f(x) dx = ∫ from a to c of f(x) dx + ∫ from c to b of f(x) dx
Riemann Sum Limit
∫ from a to b of f(x) dx = lim(n→∞) ∑ from i=1 to n of f(xᵢ*) · Δx
8Memory Aids
"Top minus Bottom, Right minus Left"
When evaluating a definite integral, find the antiderivative and compute F(b) - F(a). Think "upper limit value minus lower limit value" -- the same idea as "top minus bottom."
"Rectangles Getting Thinner"
Picture filling the area under a curve with rectangles. As the rectangles get thinner and thinner (more and more of them), the approximation becomes perfect. That perfect limit is the definite integral.
"Definite = Definite Number"
A definite integral always gives you a definite (specific) number, not a function. If your answer still has x in it, something went wrong. An indefinite integral gives a function family (with +C).
"Sign Check: Above vs. Below"
If the function dips below the x-axis, the integral of that portion is negative. If the problem asks for total area (not net area), you must take the absolute value of each piece or split at zeros and add the absolute values.
"No +C for Definite!"
When evaluating a definite integral, you never need the constant of integration. It cancels in F(b) - F(a). Save the +C for indefinite integrals only.
9Common Mistakes
Forgetting to evaluate at the lower limit
Students sometimes only evaluate the antiderivative at the upper limit and forget to subtract the evaluation at the lower limit. Always compute F(b) - F(a), not just F(b).
Confusing net signed area with total area
The definite integral gives net signed area, where regions below the x-axis are negative. If the problem asks for total area, you must split at zeros and take absolute values.
Adding +C to a definite integral
The constant of integration is only needed for indefinite integrals. For definite integrals, C always cancels out in F(b) - F(a), so omit it entirely.
Errors in finding the antiderivative
The FTC Part 2 process relies entirely on finding the correct antiderivative. Common errors include forgetting to adjust coefficients (e.g., the antiderivative of e2x is (1/2)e2x, not e2x) or incorrect power rule application.
Not splitting at discontinuities or absolute values
When integrating functions with absolute values or piecewise definitions, you must split the integral at points where the definition changes. Failing to do so leads to incorrect results.
Swapping limits without changing sign
If you need to reverse the order of integration (e.g., from ∫ from b to a to ∫ from a to b), you must negate the integral. Forgetting the sign change is a common error.
Substitution errors with Riemann sum types
When computing Riemann sums, be careful to use the correct sample points: left endpoints for LRAM, right endpoints for RRAM, and midpoints for MRAM. Mixing these up produces wrong approximations.
Quick Revision Summary
- ✓A definite integral calculates the net signed area between a function and the x-axis over a specific interval.
- ✓Unlike indefinite integrals, a definite integral produces a single numerical value, not a function.
- ✓Riemann sums approximate the area using rectangles (LRAM, RRAM, MRAM) or trapezoids. The definite integral is the limit as the number of rectangles approaches infinity.
- ✓FTC Part 2: ∫ from a to b of f(x) dx = F(b) - F(a), where F'(x) = f(x).
- ✓Key properties include order of integration, constant multiple, sum/difference, and additivity.
- ✓Area above the x-axis is positive; area below is negative. For total area, use absolute values.
- ✓No +C is needed for definite integrals -- the constant cancels in F(b) - F(a).
- ✓Split integrals at points where the function changes definition (absolute values, piecewise functions).
- ✓Swapping limits of integration changes the sign of the integral.
- ✓Always double-check your antiderivative before evaluating -- errors there cascade through the entire solution.
Frequently Asked Questions
- What is the difference between a definite and an indefinite integral?
- An indefinite integral represents a family of antiderivatives and includes a constant of integration (+C). Its result is a function. A definite integral has specific upper and lower limits of integration and produces a single numerical value representing the net signed area between the curve and the x-axis over the given interval.
- Why is it called "net signed area" instead of just "area"?
- The definite integral calculates net signed area because regions above the x-axis contribute positive values while regions below the x-axis contribute negative values. If you want the total (unsigned) area, you must integrate the absolute value of the function or split the integral at x-intercepts and take the absolute value of each piece.
- Do I need to include the constant of integration (+C) when evaluating a definite integral?
- No. When evaluating a definite integral using the Fundamental Theorem of Calculus Part 2, the constant of integration cancels out: (F(b) + C) - (F(a) + C) = F(b) - F(a). So you can always use an antiderivative with C = 0.
- Which Riemann sum method gives the best approximation?
- For a given number of subintervals, the Midpoint Riemann Sum generally provides a better approximation than Left or Right Riemann Sums. The Trapezoidal Rule typically gives an even more accurate result. However, as the number of subintervals approaches infinity, all methods converge to the exact value of the definite integral.
- What happens when I swap the limits of integration?
- Swapping the upper and lower limits of integration changes the sign of the integral. That is, the integral from a to b of f(x) dx equals the negative of the integral from b to a of f(x) dx. This is one of the fundamental properties of definite integrals.
Practice Quiz
Test your knowledge — select the correct answer for each question.
1.The definite integral ∫ₐᵇ f(x)dx represents:
2.In ∫ₐᵇ f(x)dx, a and b are called:
3.∫ₐᵇ f(x)dx = −∫ᵇₐ f(x)dx:
4.∫ₐᵇ c·f(x)dx equals:
5.If F'(x) = f(x), then ∫ₐᵇ f(x)dx =:
6.A left Riemann sum uses the function value at:
7.If f(x) ≥ 0 on [a, b], the definite integral gives:
8.∫ₐᵃ f(x)dx equals:
9.As the number of rectangles in a Riemann sum increases:
10.∫ₐᵇ [f(x) + g(x)]dx equals:
Final Study Advice
- 1.Practice evaluating definite integrals with polynomials, trig functions, and exponentials until the FTC Part 2 process becomes second nature.
- 2.Always verify your antiderivative by differentiating it to check that you get the original integrand back.
- 3.Memorize the key properties of definite integrals -- they frequently appear on AP exams as standalone questions or parts of multi-step problems.
- 4.Sketch the function before integrating to identify sign changes and decide whether to split the integral.
- 5.Connect definite integrals to real-world interpretations: displacement from velocity, total revenue from marginal revenue, accumulated quantity from rate.