Integral Calculus and the Fundamental Theorem

Introduction

Differential calculus describes instantaneous change; integral calculus describes accumulation. In mechatronics, accumulation appears whenever a quantity is obtained by adding many small contributions: position from velocity, charge from current, energy from power, mass from flow rate, or total error from a control signal. Integral calculus provides both a conceptual language for accumulation and computational tools for evaluating it.

This chapter develops antiderivatives, definite and indefinite integrals, Riemann sums, properties of integration, substitution, and the Fundamental Theorem of Calculus. Particular attention is given to interpretation, assumptions, units, and engineering applications.

Learning Objectives

By the end of this chapter, you should be able to:

1. Explain integration as accumulation and as the inverse of differentiation.
2. Approximate accumulated quantities using Riemann sums.
3. Distinguish definite and indefinite integrals.
4. Apply basic integration rules and properties.
5. Use substitution to evaluate composite-function integrals.
6. State and apply both parts of the Fundamental Theorem of Calculus.
7. Interpret integrals in mechatronics and verify answers using units and physical reasoning.

1. Accumulation and Antiderivatives

Suppose a robot moves with velocity v(t) metres per second. Over a short interval of duration ∆t, its displacement is approximately v(t)∆t. If velocity varies, the total displacement over [a,b ] is found by adding many such contributions.

An antiderivative reverses differentiation. A function F(x) is an antiderivative of f(x) on an interval if

For example, since d(x³/3)/dx = x², an antiderivative of x² is x³/3. However, it is not the only one: every function

has derivative x², where C is an arbitrary constant. The constant is essential because differentiation removes constants.

Basic antiderivative rules

For a constant k and exponent n ≠ −1,

Other important rules are:

The power rule does not apply when n = −1; that case produces a logarithm. In trigonometric formulas, angles are measured in radians.

Integration is linear:

where α and β are constants and the functions are integrable on the domain under consideration.

Worked Example 1: An antiderivative

Find all antiderivatives of f(t) = 6t² − 4t + 5.

Integrating term by term gives

Differentiation verifies the result:

If a physical initial condition is supplied, it determines C. For example, if position is s(0) = 2 metres and velocity is v(t) = 6t² − 4t + 5 metres per second, then

and s(0)=C=2. Thus s(t)=2t³−2t²+5t+2 metres.

2. Definite Integrals and Riemann Sums

A definite integral represents net accumulation over a closed interval. Let f be bounded on [a,b]]. Divide the interval into n subintervals of equal width

Choose a sample point x_i^* in each subinterval. The corresponding Riemann sum is

If these sums approach a unique limit as n → ∞, the function is Riemann integrable and

The symbol dx indicates the limiting width of the small intervals. The definite integral is a number, not a family of functions. If f(x) has units of metres per second and dx has units of seconds, then ∫ f(x)dx has units of metres.

A definite integral is signed. Regions below the horizontal axis contribute negatively. Consequently, it gives net signed area or net accumulation, not necessarily total geometric area. Total area requires integration of |f(x)| or a suitable partition at zeros of f].

Left, right, and midpoint approximations

For a partition with width ∆x, the left approximation uses left endpoints, the right approximation uses right endpoints, and the midpoint approximation uses subinterval midpoints. For smooth functions, midpoint approximations are often more accurate than left or right approximations for the same number of subintervals, although exact error depends on the function.

Worked Example 2: A Riemann-sum estimate

Approximate

using a right-endpoint sum with n=4. Here ∆t=0.5, and the right endpoints are 0.5, 1, 1.5, 2. Therefore,

The exact value, obtained later using the Fundamental Theorem, is 8/3 ≈ 2.667. The right sum overestimates because t² is increasing on [0,2]].

3. Definite and Indefinite Integrals

The indefinite integral

denotes the family of all antiderivatives of f. It is a function family and includes an arbitrary constant.

The definite integral

is a single accumulated value. It has no arbitrary constant because constants cancel when endpoint values are subtracted. The two concepts are connected by the Fundamental Theorem of Calculus.

Important properties include:

The last property holds whenever the integrals exist and is useful for splitting a process into stages.

If f(x) ≤ g(x) throughout [a,b]], then

Also, if m ≤ f(x) ≤ M on the interval, then

These inequalities provide useful engineering bounds and plausibility checks.

4. The Fundamental Theorem of Calculus

The Fundamental Theorem has two related parts.

Part I: Differentiating an accumulation function

If f is continuous on an interval containing a and x, define

Then

The variable t is a dummy variable of integration; x is the variable endpoint. This theorem states that accumulation from a fixed starting point has instantaneous rate equal to the current integrand.

More generally, if u(x) is differentiable and f is continuous, then

This is the chain rule applied to an accumulation function.

Part II: Evaluating a definite integral

If F is any antiderivative of continuous f on [a,b]], then

This result converts a limiting accumulation problem into endpoint evaluation.

Worked Example 3: Evaluation by the theorem

Evaluate

An antiderivative is F(x)=x²+x. Hence

If x is measured in seconds and 2x+1 in metres per second, the result is 10 metres.

Worked Example 4: Variable upper limit

Let

Because the integrand is continuous,

No full integration is needed to find the derivative.

5. Substitution

Substitution is the integration counterpart of the chain rule. Suppose u=g(x) is differentiable. Then du=g′(x)dx, and

For a definite integral, the limits must also be changed when the variable changes:

Alternatively, one may substitute back to x before applying the original limits, but mixing variables and limits is an error.

Worked Example 5: Indefinite substitution

Evaluate

Set u=x². Then du=2x dx, so

Worked Example 6: Definite substitution

Evaluate

Let u=x². The limits become u=0 and u=1. Therefore,

Changing the limits avoids the need to substitute back.

6. Applications in Mechatronics

Position, velocity, and acceleration

The relationships among position s(t), velocity v(t), and acceleration a(t) are

Thus,

and

Displacement is signed. Distance travelled is obtained by integrating speed |v(t)|:

Electrical charge and current

If current i(t) is measured in amperes, then charge accumulated between t₁ and t₂ is

Since one ampere equals one coulomb per second, the units are coulombs. If i(t) is constant, this reduces to ∆q=i∆t.

Energy and power

If power P(t) is measured in watts, energy is

Because one watt equals one joule per second, the result has units of joules when time is in seconds. For a motor, integrating measured electrical power estimates energy consumption; integrating mechanical power estimates delivered mechanical energy.

Sensor data and numerical integration

Real control systems often provide sampled data rather than an exact function. For samples separated by constant interval ∆t, a trapezoidal estimate is

This formula assumes equally spaced samples. For nonuniform samples, each interval must use its own width:

Sampling rate, sensor noise, bias, saturation, and numerical drift affect the result. Integration of a small constant bias can grow steadily with time, which is why inertial navigation systems require calibration and sensor fusion.

7. Illustrative Case: Estimating Robot Travel from Encoder Data

Background

Consider a mobile robot whose wheel encoder estimates forward velocity during a 4-second interval. The measured velocities are:

Problem

Estimate the forward displacement using the trapezoidal rule and assess the meaning of the result.

Analysis and solution

The sampling interval is 1 second. Applying the formula,

All velocities are nonnegative, so displacement and distance are both approximately 3.65 m. If the robot had reversed direction, signed velocity would have produced net displacement, while distance would require integrating the absolute velocity.

Engineering interpretation

The estimate is not exact because velocity between samples is approximated by straight-line segments. Accuracy can be improved by increasing the sampling rate, using calibrated sensors, and checking the result against an independent measurement such as wheel rotation or an optical position sensor. A constant encoder bias of 0.02 m/s over 4 seconds would contribute approximately 0.08 m of displacement error, illustrating how integration accumulates bias.

Learning points

1. A velocity integral produces displacement.
2. Units provide a direct validation: (m/s)×s=m.
3. Numerical integration is an approximation whose quality depends on sampling and signal quality.
4. Net displacement and total distance are different quantities.

8. Common Errors, Limitations, and Best Practices

Confusing area with signed accumulation

An integral below the axis is negative. Do not report it as positive area unless the problem specifically requests geometric area or distance.

Omitting the constant of integration

The constant C must appear in an indefinite integral. It is determined by an initial or boundary condition.

Mishandling substitution limits

When changing from x to u, either change the limits or substitute back before evaluating. Do not use old limits with the new variable.

Ignoring continuity and integrability

The Fundamental Theorem applies directly when the integrand is continuous on the relevant interval. Some discontinuous functions remain integrable, but improper integrals and singularities require separate analysis.

Ignoring units and initial conditions

An integral cannot be physically interpreted without identifying the variable of integration and its units. Initial conditions are necessary to reconstruct an absolute state from a rate.

Numerical limitations

Numerical integration may suffer from truncation error, round-off error, noisy measurements, nonuniform sampling, and accumulated bias. In embedded systems, engineers should document the integration method, sampling period, numerical precision, and reset or correction strategy.

Chapter Summary

Integral calculus formalizes accumulation. An antiderivative reverses differentiation and is determined up to an arbitrary constant. A definite integral is the limit of Riemann sums and represents signed accumulation over an interval. Its properties include linearity, interval splitting, reversal of limits, and order preservation.

The Fundamental Theorem of Calculus connects the two viewpoints: differentiating an accumulation function returns the integrand, and evaluating a definite integral requires subtracting antiderivative values at the endpoints. Substitution extends integration to composite functions and follows from the chain rule.

In mechatronics, integration converts velocity to displacement, acceleration to velocity change, current to charge, and power to energy. Real systems use numerical methods such as the trapezoidal rule, where sampling quality, sensor bias, units, and initial conditions must be considered carefully.

Key Terms

Antiderivative: A function whose derivative equals a specified function.

Accumulation: The total obtained by adding contributions over an interval.

Definite integral: A signed accumulated value between two bounds.

Indefinite integral: The family of all antiderivatives, including an arbitrary constant.

Riemann sum: A finite sum used to approximate an integral.

Partition: A division of an interval into subintervals.

Integrand: The function being integrated.

Dummy variable: A variable, such as t, whose name may be changed without altering an integral.

Net displacement: Final position minus initial position, obtained by integrating signed velocity.

Distance travelled: The integral of speed or absolute velocity.

Fundamental Theorem of Calculus: The theorem connecting differentiation and definite integration.

Substitution: A change of variable used to simplify an integral.

Continuity: The property that a function has no breaks or jumps at a point or interval.

Trapezoidal rule: A numerical integration method based on trapezoid-shaped approximations.

Sampling interval: The time or spatial separation between successive measurements.

Bias: A persistent systematic measurement error.

Initial condition: A known value used to determine an integration constant.

Signed area: Accumulation in which values below the axis contribute negatively.

Review Questions

5 Short Answer Questions

1. What is an antiderivative, and why does an indefinite integral include C?
2. Define a Riemann sum and explain what happens as the number of subintervals increases.
3. State the two parts of the Fundamental Theorem of Calculus.
4. What is the difference between displacement and distance travelled?
5. Why must limits be changed when substitution is used directly in a definite integral?

5 Long Answer Questions

1. Explain definite and indefinite integrals, including their notation, interpretation, and engineering units.
2. Derive the relationship between position, velocity, and acceleration through integration, including the role of initial conditions.
3. Explain the Fundamental Theorem of Calculus and illustrate both parts with suitable examples.
4. Compare left-endpoint, right-endpoint, midpoint, and trapezoidal approximations in terms of interpretation and practical use.
5. Discuss the effects of sampling rate, sensor noise, and bias when integrating measured mechatronic signals.

3 Analytical / Application Questions

1. A motor shaft has angular velocity ω(t)=4t−t² rad/s for 0≤t≤3. Determine its angular displacement and explain whether angular distance equals angular displacement.
2. A current sensor records i(t)=2+0.5t amperes for 6 seconds. Calculate the charge delivered and verify the units.
3. A robot's acceleration is measured numerically over a fixed interval. Design a calculation procedure for estimating velocity change using sampled data, and identify two sources of error and one method of correction.

Multiple Choice Questions with Answers

Question 1

Which expression represents all antiderivatives of 6x²?

A. 2x³+C

B. 6x³+C

C. 3x²+C

D. 2x²+C

Correct Answer: A. 2x³+C

Question 2

What is the exact value of ∫₀² x dx?

A. 1

B. 2

C. 3

D. 4

Correct Answer: B. 2

Question 3

If F′(x)=f(x), which expression equals ∫ₐᵇ f(x)dx?

A. F(a)+F(b)

B. F(a)−F(b)

C. F(b)−F(a)

D. F′(b)−F′(a)

Correct Answer: C. F(b)−F(a)

Question 4

If G(x)=∫₁ˣ (t²+3)dt, what is G′(x)?

A. 2x+3

B. x²+3

C. x³/3+3x

D. 1²+3

Correct Answer: B. x²+3

Question 5

Evaluate ∫ 2x cos(x²)dx.

A. cos(x²)+C

B. 2sin(x²)+C

C. sin(x²)+C

D. x² sin(x²)+C

Correct Answer: C. sin(x²)+C

Question 6

If velocity is measured in metres per second, what are the units of ∫ v(t)dt?

A. Metres

B. Metres per second squared

C. Seconds per metre

D. Metres squared

Correct Answer: A. Metres

Question 7

For f(x)≥0 on [a,b], what does ∫ₐᵇ f(x)dx represent?

A. Only the maximum value of f

B. The signed area, which is nonnegative here

C. The derivative of f

D. The average value without multiplying by interval length

Correct Answer: B. The signed area, which is nonnegative here

Question 8

Using a right-endpoint sum with n=2, what is the approximation to ∫₀² x²dx?

A. 1

B. 2

C. 4

D. 5

Correct Answer: C. 4

Question 9

Which substitution is most suitable for ∫ 3x² e^x³dx?

A. u=3x²

B. u=x²

C. u=e^x³

D. u=x³

Correct Answer: D. u=x³

Question 10

If current is integrated with respect to time, the result is electrical:

A. Power

B. Resistance

C. Charge

D. Voltage

Correct Answer: C. Charge