ECON1202 Chap.11 Integration and Differential Equations
Integration and Differential Equations
Integration reverses differentiation and represents accumulated change. An indefinite integral is a family of antiderivatives and includes an arbitrary constant. A definite integral evaluates an antiderivative at the upper and lower bounds and subtracts, producing a signed net accumulation. Constants factor out and sums split term by term.
The power rule applies except to one over x, whose antiderivative is the logarithm of absolute x. Exponential antiderivatives retain their exponential shape with an adjustment for the exponent coefficient. Differential equations specify how a variable changes rather than giving its level directly.
Exponential growth makes the rate proportional to the current stock, producing a solution based on the initial value and an exponential factor. Limited growth makes the rate proportional to the remaining gap from a capacity, so the solution approaches that capacity as the gap decays.
Proposed solutions should be checked twice: substitute the initial time to recover the initial value, then differentiate to recover the stated rate law. Interpretation distinguishes reaching a level from approaching it asymptotically.
What this chapter covers
- 01
Antiderivative families
- 02
Power-rule integration
- 03
Exponential and logarithmic integrals
- 04
Linearity of integration
- 05
Definite integrals
- 06
Exponential growth equations
- 07
Limited growth equations
- 08
Initial-condition verification
Worked example · free
Solve and verify a limited-growth model
- +1Match the limited-growth form with M=800, k=0.25 and y0=200.
- +1Compute A=M-y0=600 and write y(t)=800-600e^(-0.25t).
- +1Evaluate y(4)=800-600e^(-1), approximately 579.27.
- +1Check y(0)=200 and confirm the solution approaches 800 from below.
Key terms
- Antiderivative
- A function whose derivative equals the specified integrand.
- Definite integral
- A signed accumulated value found by evaluating an antiderivative across fixed bounds.
- Integration constant
- The arbitrary additive constant required for an indefinite antiderivative family.
- Differential equation
- An equation linking a variable to one or more of its rates of change.
- Exponential growth
- A model where the rate of change is proportional to the current level.
- Limited growth
- A model where the rate of change is proportional to the remaining gap from a capacity.
Integration and Differential Equations FAQ
When is an integration constant needed?
Include it for an indefinite integral because many functions share the same derivative. In a definite integral the constant cancels between upper and lower evaluations, so the final numeric accumulation does not depend on it.
What does a definite integral represent?
It represents signed net accumulation over the interval. Regions where the integrand is negative subtract from the result. A request for total geometric area may require splitting at zeros and adding positive magnitudes instead.
How do exponential and limited growth differ?
Exponential growth follows the current stock, so the rate can accelerate as the level rises. Limited growth follows the remaining gap to a capacity, so the rate slows as the level approaches that ceiling.
How should a differential-equation solution be verified?
Substitute the initial time to check the constant, then differentiate the proposed function and compare it with the right-hand side of the rate law. Also inspect the long-run direction and units of the rate parameter.
How can I keep integration and differential-equation working auditable?
For an indefinite integral, show the antiderivative family and include the constant, then differentiate it to recover the integrand. For a definite integral, write F(b)-F(a) with the entire lower-bound value bracketed, report accumulated units and use the sign of the integrand as a plausibility check. When recovering a total function from a marginal one, apply the known initial level after integrating.
For a differential equation, identify whether the rate is proportional to the current stock or to a remaining gap. Separate variables or use the stated solution pattern, then apply the initial condition to select one trajectory. Verify both the initial value and the rate law by differentiation.
Finally inspect long-run behaviour: exponential growth or decay follows the sign of its proportional rate, while a limited-growth model should move toward its stated ceiling under the basic assumptions. Keep units on the rate parameter: its reciprocal indicates a time scale, which helps identify an exponent using the wrong time unit.
Exam move
Build an integration table from the formulas printed on the course sheet, but attach the condition to each entry, including the special one-over-x case. Practise indefinite and definite versions side by side so the role of the constant and bounds remains clear. Use parentheses when evaluating F(b)-F(a), especially with negative lower bounds. Differentiate every antiderivative as a routine check.
For differential equations, classify the right-hand side before selecting a solution family: current stock indicates exponential growth, while remaining gap indicates limited growth. Determine the constant from the initial condition, then verify both initial value and derivative. Sketch the expected path and long-run level so a sign error in the exponent becomes visually obvious.
Keep time units consistent with k and use approaches rather than reaches when the solution tends asymptotically to a capacity.
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