The Central Idea
A definite integral records exact signed accumulation over an interval.
\[ \boxed{\int_a^b f(x)\,dx} \]It is defined as the limit of increasingly fine Riemann sums:
\[ \boxed{\int_a^b f(x)\,dx=\lim\sum f(x_i^*)\Delta x_i}. \]The basic movement is
\[ \boxed{\text{finite approximation}\to\text{limit}\to\text{exact accumulation}}. \]Parts of the Notation
In
\[ \int_a^b f(x)\,dx, \]- \(\int\) is the integral sign;
- \(a\) is the lower limit of integration;
- \(b\) is the upper limit of integration;
- \(f(x)\) is the integrand;
- \(x\) is the variable of integration;
- \(dx\) indicates accumulation with respect to \(x\).
The Limits of Integration
The values \(a\) and \(b\) specify the interval over which accumulation is measured.
Thus
\[ \int_2^7 f(x)\,dx \]means to accumulate the contributions associated with \(f(x)\) as \(x\) runs from \(2\) to \(7\).
Definition Through Riemann Sums
Partition \([a,b]\) into subintervals of widths \(\Delta x_i\), and choose a representative point \(x_i^*\) in each subinterval.
Form
\[ \sum_{i=1}^{n}f(x_i^*)\Delta x_i. \]If these sums approach a unique value as the largest subinterval width approaches zero, then
\[ \boxed{\int_a^b f(x)\,dx=\lim_{\|P\|\to0}\sum_{i=1}^{n}f(x_i^*)\Delta x_i}. \]Equal-Width Version
For \(n\) equal subintervals,
\[ \Delta x=\frac{b-a}{n}. \]A common version of the definition is
\[ \boxed{\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^{n}f(x_i^*)\Delta x}. \]Integrable Functions
A function is Riemann integrable on \([a,b]\) when its Riemann sums approach a unique limit as the partition is refined.
Every continuous function on a closed interval is Riemann integrable. Many functions with a finite number of simple discontinuities are also integrable.
Geometric Interpretation
If \(f(x)\ge0\) on \([a,b]\), then
\[ \int_a^b f(x)\,dx \]equals the geometric area between the graph of \(f\), the \(x\)-axis, and the lines \(x=a\) and \(x=b\).
\[ \boxed{\text{definite integral}=\text{area under the curve}} \]when the function remains nonnegative.
Signed Area
The definite integral is fundamentally a measure of signed area.
- Regions above the \(x\)-axis contribute positively.
- Regions below the \(x\)-axis contribute negatively.
When the Function Is Negative
If \(f(x)\le0\) on \([a,b]\), the geometric region has positive area magnitude, but the definite integral is negative.
If the region has geometric area \(A\), then
\[ \boxed{\int_a^b f(x)\,dx=-A}. \]Positive and Negative Contributions Can Cancel
If one part contributes \(+6\) and another contributes \(-6\), then
\[ \int_a^b f(x)\,dx=0. \]A zero integral does not necessarily mean that no geometric region exists. It may mean that positive and negative contributions cancel.
Total Geometric Area
If total geometric area is required, negative regions must be counted by magnitude.
\[ \boxed{\text{total area}=\int_a^b |f(x)|\,dx}. \]In practice, one often splits the interval at zeros of \(f\) and adds the positive magnitudes of each region.
Example: Constant Function
Evaluate geometrically:
\[ \int_1^5 3\,dx. \]The graph forms a rectangle of height \(3\) and width \(4\).
\[ \boxed{\int_1^5 3\,dx=3(4)=12}. \]Example: Constant Below the Axis
Evaluate:
\[ \int_2^6(-4)\,dx. \]The rectangle has geometric area \(4(6-2)=16\), but it lies below the axis.
\[ \boxed{\int_2^6(-4)\,dx=-16}. \]Example: Triangle
Suppose \(f\) is linear on \([0,4]\), with \(f(0)=0\) and \(f(4)=6\).
The region is a triangle of base \(4\) and height \(6\):
\[ \int_0^4 f(x)\,dx=\frac12(4)(6). \] \[ \boxed{12}. \]Example: Semicircle Geometry
Suppose the graph on \([-3,3]\) is the upper semicircle of radius \(3\).
The full circle has area \(9\pi\), so the upper half has area
\[ \boxed{\int_{-3}^{3} f(x)\,dx=\frac{9\pi}{2}}. \]Orientation Matters
Reversing the limits changes the sign:
\[ \boxed{\int_a^b f(x)\,dx=-\int_b^a f(x)\,dx}. \]A definite integral is therefore an oriented accumulation.
Example: Reversing the Limits
If
\[ \int_2^7 f(x)\,dx=11, \]then
\[ \boxed{\int_7^2 f(x)\,dx=-11}. \]Equal Limits Give Zero
If the interval begins and ends at the same point, there is no interval length over which to accumulate:
\[ \boxed{\int_a^a f(x)\,dx=0}. \]Additivity Across Adjacent Intervals
If \(c\) lies between \(a\) and \(b\), then
\[ \boxed{\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx}. \]Example: Combining Intervals
Suppose
\[ \int_1^4f(x)\,dx=7 \]and
\[ \int_4^9f(x)\,dx=-2. \]Then
\[ \boxed{\int_1^9f(x)\,dx=7+(-2)=5}. \]Example: Solving for a Missing Interval
Suppose
\[ \int_0^8f(x)\,dx=13, \qquad \int_0^3f(x)\,dx=5. \]Since
\[ \int_0^8 f=\int_0^3 f+\int_3^8 f, \]we get
\[ \boxed{\int_3^8 f(x)\,dx=8}. \]Constant Multiple Property
\[ \boxed{\int_a^b c f(x)\,dx=c\int_a^b f(x)\,dx}. \]If \(\int_1^5f(x)\,dx=4\), then
\[ \int_1^5 3f(x)\,dx=12. \]Sums and Differences
\[ \boxed{\int_a^b[f(x)+g(x)]\,dx=\int_a^b f(x)\,dx+\int_a^b g(x)\,dx} \]and
\[ \boxed{\int_a^b[f(x)-g(x)]\,dx=\int_a^b f(x)\,dx-\int_a^b g(x)\,dx}. \]Linearity of the Integral
Combining the preceding rules gives
\[ \boxed{\int_a^b[\alpha f(x)+\beta g(x)]\,dx=\alpha\int_a^b f(x)\,dx+\beta\int_a^b g(x)\,dx}. \]Example: Using Linearity
Suppose
\[ \int_0^4f(x)\,dx=6, \qquad \int_0^4g(x)\,dx=-3. \]Then
\[ \int_0^4[2f(x)-5g(x)]\,dx=2(6)-5(-3)=27. \] \[ \boxed{27}. \]Integral of a Constant
For any constant \(c\),
\[ \boxed{\int_a^b c\,dx=c(b-a)}. \]Order Property
If
\[ f(x)\le g(x) \]for every \(x\in[a,b]\), then
\[ \boxed{\int_a^b f(x)\,dx\le\int_a^b g(x)\,dx}. \]Positivity
If \(f(x)\ge0\) on \([a,b]\), then
\[ \boxed{\int_a^b f(x)\,dx\ge0}. \]Bounding an Integral
If
\[ m\le f(x)\le M \]on \([a,b]\), where \(a
An Absolute-Value Bound
If \(|f(x)|\le M\) on \([a,b]\), then
\[ \boxed{\left|\int_a^b f(x)\,dx\right|\le M(b-a)}. \]Net Accumulation versus Total Accumulation
In general,
\[ \boxed{\left|\int_a^b f(x)\,dx\right|\le\int_a^b |f(x)|\,dx}. \]The left side is the magnitude of net accumulation. The right side counts all contribution magnitudes without cancellation.
Symmetry: Odd Functions
If \(f\) is odd, so that \(f(-x)=-f(x)\), then
\[ \boxed{\int_{-a}^{a}f(x)\,dx=0}. \]Symmetry: Even Functions
If \(f\) is even, so that \(f(-x)=f(x)\), then
\[ \boxed{\int_{-a}^{a}f(x)\,dx=2\int_0^a f(x)\,dx}. \]Example: Odd Symmetry
Since \(x^3\) is odd,
\[ \boxed{\int_{-5}^{5}x^3\,dx=0}. \]Example: Even Symmetry
Since \(x^2\) is even,
\[ \int_{-3}^{3}x^2\,dx=2\int_0^3x^2\,dx. \]Units of a Definite Integral
If \(f(x)\) has units \(A/B\) and \(dx\) has units \(B\), then
\[ f(x)\,dx \]has units
\[ \boxed{A}. \]Units multiply exactly as they did in the Riemann-sum interpretation.
Velocity and Displacement
If \(v(t)\) is velocity, then
\[ \boxed{\int_{t_1}^{t_2}v(t)\,dt} \]gives displacement. Positive velocity contributes positively; negative velocity contributes negatively.
Displacement versus Total Distance
Net displacement is
\[ \int_a^b v(t)\,dt. \]Total distance is
\[ \boxed{\int_a^b |v(t)|\,dt}. \]Flow Rate and Accumulated Volume
If \(r(t)\) is measured in liters per minute, then
\[ \int_a^b r(t)\,dt \]has units of liters and represents net volume accumulated.
Density and Mass
If \(\rho(x)\) is linear density, then
\[ \boxed{\int_a^b \rho(x)\,dx} \]gives total mass across the interval.
Marginal Quantities
If \(C'(q)\) is marginal cost, then
\[ \boxed{\int_{q_1}^{q_2}C'(q)\,dq} \]gives the change in total cost as production moves from \(q_1\) to \(q_2\).
Probability Density
If \(f(x)\) is a probability density function, then
\[ \boxed{\int_a^b f(x)\,dx} \]represents the probability that the random variable lies between \(a\) and \(b\).
The Variable of Integration Is Local
The expressions
\[ \int_a^b f(x)\,dx \]and
\[ \int_a^b f(t)\,dt \]represent the same value if the function expression is renamed consistently. The variable of integration is local to that integral.
Definite versus Indefinite Integrals
An indefinite integral produces a family of functions:
\[ \int f(x)\,dx=F(x)+C. \]A definite integral produces a single numerical value:
\[ \int_a^b f(x)\,dx. \]Why a Definite Integral Does Not Need \(+C\)
A definite integral is a number determined by an integrand and oriented interval. It is not a family of antiderivatives.
Therefore, we do not append \(+C\) to a final definite-integral value.
The Coming Connection to Antiderivatives
Definite integrals come from limits of Riemann sums. Antiderivatives come from reversing differentiation.
The Fundamental Theorem of Calculus will show that these ideas are deeply connected.
Example: Combining Signed Regions
Suppose a graph has:
- area \(8\) above the axis on \([0,2]\);
- area \(3\) below the axis on \([2,5]\);
- area \(4\) above the axis on \([5,7]\).
Then
\[ \int_0^7f(x)\,dx=8-3+4. \] \[ \boxed{9}. \]Example: Total Area from the Same Regions
Total geometric area adds magnitudes:
\[ 8+3+4=15. \] \[ \boxed{\text{total area}=15}. \]Example: Algebra with Known Integral Values
Suppose
\[ \int_0^6f(x)\,dx=10, \qquad \int_0^6g(x)\,dx=4. \]Evaluate
\[ \int_0^6[3f(x)+2g(x)-5]\,dx. \]By linearity,
\[ 3(10)+2(4)-5(6)=8. \] \[ \boxed{8}. \]Example: Reversed and Split Intervals
Suppose
\[ \int_1^3f(x)\,dx=4, \qquad \int_3^6f(x)\,dx=7. \]Then
\[ \int_1^6f(x)\,dx=11 \]and therefore
\[ \boxed{\int_6^1f(x)\,dx=-11}. \]Definite Integrals Can Be Estimated Numerically
When exact symbolic evaluation is difficult or unavailable, a definite integral can still be approximated.
- left and right Riemann sums;
- midpoint sums;
- trapezoidal approximations;
- Simpson-type methods;
- computer-based quadrature.
These are developed later in Numerical Integration.
Common Error: Treating the Integral as Ordinary Area
Regions below the axis contribute negatively. A definite integral is net signed accumulation, not automatically total positive area.
Common Error: Reversing Limits Without Changing Sign
\[ \boxed{\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx}. \]Common Error: Adding \(+C\)
A definite integral evaluates to a number. A result such as \(12+C\) is not an appropriate final definite-integral value.
Common Error: Splitting Intervals with the Wrong Orientation
The correct rule is
\[ \int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx. \]Each piece must preserve the intended direction.
Common Error: Taking Absolute Value Too Early
If the problem asks for the definite integral, preserve the sign of every contribution. Use absolute value only when total magnitude or total geometric area is intended.
Common Error: Ignoring Units
If a rate is measured in units per time and we integrate with respect to time, the result has units of the accumulated quantity.
\[ \frac{\text{meters}}{\text{second}}\cdot\text{seconds}=\text{meters}. \]A Reliable Procedure for Geometric Evaluation
- Inspect or sketch the graph.
- Split the interval at axis crossings or other important points.
- Identify familiar geometric regions.
- Compute each region's area magnitude.
- Use positive signs above the axis and negative signs below it.
- Add the signed contributions.
- If total area is requested, add magnitudes instead.
A Reliable Procedure with Given Integral Values
- Identify interval and orientation.
- Split or combine intervals using additivity.
- Reverse limits carefully when needed.
- Use linearity for sums and constant multiples.
- Evaluate constant pieces using \(c(b-a)\).
- Substitute known integral values.
- Simplify after the structure is correct.
Practice 1: Rectangle
Evaluate geometrically:
\[ \int_2^7 4\,dx. \]Practice 2: Negative Rectangle
Evaluate:
\[ \int_{-1}^{3}(-5)\,dx. \]Practice 3: Reverse Limits
If
\[ \int_1^8f(x)\,dx=14, \]find
\[ \int_8^1f(x)\,dx. \]Practice 4: Split an Interval
Suppose
\[ \int_0^5f(x)\,dx=9, \qquad \int_0^2f(x)\,dx=4. \]Find
\[ \int_2^5f(x)\,dx. \]Practice 5: Combine Intervals
Suppose
\[ \int_2^6f(x)\,dx=-3, \qquad \int_6^{10}f(x)\,dx=8. \]Find
\[ \int_2^{10}f(x)\,dx. \]Practice 6: Linearity
Suppose
\[ \int_0^3f(x)\,dx=5, \qquad \int_0^3g(x)\,dx=-2. \]Evaluate
\[ \int_0^3[4f(x)+3g(x)]\,dx. \]Practice 7: Constant Term
Suppose
\[ \int_1^4f(x)\,dx=7. \]Evaluate
\[ \int_1^4[2f(x)-3]\,dx. \]Practice 8: Odd Symmetry
Evaluate without finding an antiderivative:
\[ \int_{-6}^{6}x^5\,dx. \]Practice 9: Even Symmetry
Rewrite using symmetry:
\[ \int_{-4}^{4}(x^2+1)\,dx. \]Practice 10: Signed Geometry
A graph encloses area \(7\) above the axis on \([0,3]\) and area \(5\) below the axis on \([3,6]\).
Find
\[ \int_0^6f(x)\,dx. \]Practice 11: Total Area
Using the same regions as Practice 10, find the total geometric area between the graph and the \(x\)-axis.
Practice 12: Units and Interpretation
A flow rate \(r(t)\) is measured in gallons per minute. What does
\[ \int_0^{10}r(t)\,dt \]represent, and what are its units?
Practice 13: Identify the Error
A student is told that
\[ \int_2^5f(x)\,dx=9 \]and concludes that
\[ \int_5^2f(x)\,dx=9. \]Explain and correct the error.
Practice 14: Definite versus Indefinite
Explain the difference between
\[ \int f(x)\,dx \]and
\[ \int_a^b f(x)\,dx. \]Practice Answers
Practice 1
\[ \boxed{\int_2^7 4\,dx=4(7-2)=20}. \]Practice 2
\[ \int_{-1}^{3}(-5)\,dx=-5(4)=\boxed{-20}. \]Practice 3
\[ \boxed{\int_8^1f(x)\,dx=-14}. \]Practice 4
\[ 9=4+\int_2^5f(x)\,dx. \] \[ \boxed{\int_2^5f(x)\,dx=5}. \]Practice 5
\[ \boxed{\int_2^{10}f(x)\,dx=-3+8=5}. \]Practice 6
\[ 4(5)+3(-2)=20-6=\boxed{14}. \]Practice 7
\[ \int_1^4[2f(x)-3]\,dx=2(7)-3(4-1)=14-9=\boxed{5}. \]Practice 8
Because \(x^5\) is odd and the interval is symmetric,
\[ \boxed{\int_{-6}^{6}x^5\,dx=0}. \]Practice 9
Because \(x^2+1\) is even,
\[ \boxed{\int_{-4}^{4}(x^2+1)\,dx=2\int_0^4(x^2+1)\,dx}. \]Practice 10
\[ \boxed{\int_0^6f(x)\,dx=7-5=2}. \]Practice 11
\[ \boxed{7+5=12}. \]Practice 12
The integral represents net accumulated volume during the first \(10\) minutes.
\[ \frac{\text{gallons}}{\text{minute}}\cdot\text{minutes}=\boxed{\text{gallons}}. \]Practice 13
Reversing the limits changes the sign:
\[ \boxed{\int_5^2f(x)\,dx=-9}. \]Practice 14
The indefinite integral \(\int f(x)\,dx\) represents a family of antiderivatives \(F(x)+C\).
The definite integral \(\int_a^b f(x)\,dx\) represents a single numerical value: net signed accumulation from \(a\) to \(b\).
What to Remember
The definite integral
\[ \boxed{\int_a^b f(x)\,dx} \]represents exact signed accumulation over an interval.
It is defined through Riemann sums:
\[ \boxed{\int_a^b f(x)\,dx=\lim\sum f(x_i^*)\Delta x_i}. \]Core interval rules are
\[ \boxed{\int_a^a f(x)\,dx=0}, \] \[ \boxed{\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx}, \]and
\[ \boxed{\int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx}. \]Most importantly:
a definite integral is an oriented total built from continuously varying local contributions.
Where We Go Next
Riemann sums define definite integrals as limits of accumulation. Antiderivatives reverse differentiation.
The next lesson, The Fundamental Theorem of Calculus, connects these ideas. It shows that accumulation can create an antiderivative and that antiderivatives provide a powerful way to evaluate definite integrals.