3.4 Application of Integration
Integration is a branch of calculus with wide applications in daily life, science, and industry. By using definite and indefinite integration, we can find areas under curves, calculate volumes of revolution for real-world objects, determine quantities from given rates of change, and solve problems in fields such as engineering, economics, and biology.
Learning Objectives
- Apply integration to calculate the internal volume of three-dimensional objects formed by revolving a curve.
- Formulate and evaluate definite integrals from given rate-of-change functions, such as bacterial colony growth rates or production rates.
- Apply the four-step mathematical problem-solving model (Understanding the problem, Planning the strategy, Implementing the strategy, Check and reflect) to solve real-world integration problems.
The 4-Step Mathematical Problem-Solving Cycle in Integration
Illustrate how real-world problems are broken down, translated into calculus operations, and verified.
From Real Object to Verified Solution
Identify dimensions / rate equations from real object.
Choose integration method: \(\int \pi x^2 \, dy\) for volume, \(\int \frac{dA}{dt} \, dt\) for total change.
Evaluate definite integral and apply scale factors (e.g., cell density or unit cost).
Check solution by reversing the equation.
Short Explanation
Integration serves as an accumulation tool for real-world quantities.
1. Areas of Regions Bounded by Curves
Definite integrals calculate the area between a curve and the \(x\)-axis or between two curves.
2. Distance from Velocity Function
Integrating a velocity function over time gives the total distance moved by an object.
3. Rate-of-Change Problems
Used in economics, biology, statistics, and engineering. Given \(\frac{dA}{dt}\) or \(\frac{dK}{dt}\), integrate over an interval to find the net accumulated quantity.
4. Generated Volumes of Revolution
For a curved profile such as \(y = ax^2\), the internal volume is found by revolving the curve around an axis.
| Application | Integral Form | Meaning |
|---|---|---|
| Volume of revolution about \(y\)-axis | \(\int_{a}^{b} \pi x^2 \, dy\) | Internal volume of a 3D object with curved profile. |
| Accumulated change from rate | \(\int_{t_1}^{t_2} \frac{dA}{dt} \, dt\) | Net change in quantity over time interval. |
| Total production | \(\int_{t_1}^{t_2} \frac{dK}{dt} \, dt\) | Total quantity produced by a machine. |
Four-Step Problem-Solving Framework
Understanding the problem: Identify given variables, constants, boundaries, units, and what needs to be determined.
Planning the strategy: Formulate equations, identify coordinates of boundary points, select appropriate integration formulas.
Implementing the strategy: Substitute known values, set up definite integrals with proper limits, integrate, evaluate, and compute final quantities with proper units.
Check and reflect: Verify the result using an alternative method, such as solving backward for the initial parameters.
Internal volume of revolution about \(y\)-axis
\(V = \int_{a}^{b} \pi x^2 \, dy\)Express \(x^2\) in terms of \(y\) before integrating.
Accumulated quantity from rate of change
\(A = \int_{t_1}^{t_2} \frac{dA}{dt} \, dt\)Integrate the rate function over the time interval.
Try to Answer First
Answer in your mind, then press "Check Answer".
The cross-section of a container is represented by \(y = ax^2\). If the diameter is \(12\text{ cm}\) when the depth is \(2\text{ cm}\), what is the value of \(a\)?
If the area of a bacterial culture grows at the rate \(\frac{dA}{dt} = 2t + 5\text{ cm}^2/\text{s}\), what is the total area formed from \(t = 0\) to \(t = 5\text{ seconds}\)?
Using the area of \(50\text{ cm}^2\) from Question 2, find the number of bacteria cells present if the cell density is \(1\,000\,000\text{ cells/cm}^2\).
Common Mistakes
- Using diameter instead of radius when substituting coordinates into equations like \(y = ax^2\) (e.g., using \(x = 12\) instead of \(x = 6\)).
- Forgetting \(\pi\) when calculating the volume of revolution of a parabolic cross-section.
- Integrating with respect to the wrong variable, such as using \(dx\) instead of \(dy\) when revolving around the \(y\)-axis.
- Neglecting the constant factor or density multiplier at the end of an accumulation problem (e.g., finding only the area \(50\text{ cm}^2\) and omitting the multiplication by \(1\,000\,000\text{ cells/cm}^2\)).
Concept Misunderstandings
Thinking that the derivative gives total quantity, rather than the instantaneous rate of change.
The derivative gives the instantaneous rate of change; to find the accumulated total, integration must be applied.
Assuming the limits of integration for volume must always start at the origin on the \(x\)-axis.
For rotation about the \(y\)-axis, the limits must be taken along the \(y\)-axis (depth values).
Believing that the "Check and reflect" stage is merely re-reading the question.
It requires active algebraic or numerical verification, such as solving backward.
Summary
- Applications of integration involve translating a physical or biological scenario into a calculus model.
- By setting up a definite integral of a rate function over time, we obtain the net change in a quantity.
- By setting up a definite integral of \(\pi x^2 \, dy\) over the vertical limits of a curved cross-section, we calculate the internal volume of three-dimensional symmetrical objects.
- The four-step problem-solving model provides a structured approach: understand, plan, implement, and check/reflect.
Short Activity
Objective: Apply integration to volumes, rates of change, and the four-step problem-solving model.
A. Objective Quiz
1 The cross-section of a bowl is modelled by \(y = \frac{1}{18}x^2\). What is the internal volume of the bowl, in \(\text{cm}^3\), if its depth is \(2\text{ cm}\)?
2 The rate of change of the area of a bacterial culture is given by \(\frac{dA}{dt} = 2t + 5\). What is the increase in area from \(t = 0\) to \(t = 5\text{ seconds}\)?
B. Fill in the Blanks
3 If a parabolic bowl has depth \(2\text{ cm}\) and diameter \(12\text{ cm}\) with equation \(y = ax^2\), the value of the constant \(a\) is ____.
4 When finding the total accumulated change of a quantity from its rate of change \(\frac{dA}{dt}\) over the interval \(t = 0\) to \(t = 5\), the mathematical operation used is ____ integration.
C. Matching / Drag and Drop
Match each step of the four-step problem-solving model or integration concept to its corresponding description or example from the textbook. Drag each term to the correct meaning. On a phone, tap a term first, then tap the matching box.
