DPST1013 Chap.9 Integration and Improper Integrals
Integration and Improper Integrals
Integration begins with adding small contributions. A rate multiplied by a short interval gives an approximate contribution over that interval; summing such products and refining the partition leads to a definite integral. For geometric graphs, the same construction uses rectangles to approximate area. The integral preserves the sign of the function, so a region below the axis subtracts from the accumulated total.
Total geometric area instead requires treating every region positively. The fundamental theorems connect this limiting sum with derivatives. One direction differentiates an accumulation function and recovers the integrand, with a chain factor when the endpoint itself changes nonlinearly. The other direction evaluates an integral by subtracting endpoint values of an antiderivative.
These relationships turn an apparently infinite summation problem into a practical calculation, but their hypotheses and endpoint conventions still matter. Substitution reverses the chain rule, while integration by parts reverses the product rule. Choosing between them is a structural decision. A visible inner expression accompanied by its derivative suggests substitution.
A product whose one factor simplifies on differentiation may favour parts. Neither method is justified by the appearance of a product alone. After finding a primitive, differentiating it checks the entire integrand, including scale factors and signs. Improper integration adds a separate question: does the accumulated quantity approach a finite value over an unbounded interval?
A finite formula for the primitive does not answer that question by itself. Introduce a finite endpoint and take a limit, or use an appropriate comparison argument. Clearly distinguish showing convergence from finding an exact value. The two conclusions require different amounts of information.
What this chapter covers
- 01
Riemann sums and bounds: multiply each representative height by its subinterval width. For increasing functions, left sums are lower and right sums are upper; for decreasing functions the roles reverse. General lower and upper sums use actual extrema on each subinterval.
- 02
Signed accumulation and geometric area: split at relevant zeros when the desired quantity counts every region positively. For area between curves, determine which curve is above on each subinterval rather than taking one final absolute value after cancellation.
- 03
Fundamental theorems: differentiate an integral-defined function by evaluating its integrand at the changing endpoint and multiplying by the endpoint derivative. Evaluate an ordinary definite integral using an antiderivative and upper-minus-lower endpoint substitution.
- 04
Substitution and parts: a substitution changes the integrand, differential and bounds consistently. Integration by parts trades the derivative of one factor for an integral of the other, and is useful only when the remaining integral becomes more manageable.
- 05
Improper integrals and comparison: define an unbounded integral by a finite-endpoint limit. Positive finite limit comparison transfers convergence behaviour from a reference tail, while two-sided infinite domains require separate convergence of both tails.
Evaluate an integral and check its scale
- 2Choose u=2+x³, giving du=3x² dx. This differential accounts for exactly the factor outside the squared bracket. The old endpoints zero and one become u=2 and u=3, so all parts of the transformed integral use the same variable.
- 2The new integral is u² from two to three. Its primitive is u³/3, so the endpoint difference is (27−8)/3=19/3. Keep the transformed bounds until evaluation is complete; substituting the old bounds here would evaluate a different integral.
- 2Returning to the original variable gives primitive (2+x³)³/3. Differentiation multiplies the squared bracket by 3x² and returns the original integrand. Evaluation at one and zero yields nine minus eight thirds, again nineteen thirds. Positivity of the integrand supports the positive result.
Key terms
- Riemann sum
- A sum of representative function values multiplied by subinterval widths, used to approximate an integral before taking a suitable limit.
- Signed area
- An area accounting convention in which contributions below the axis are negative and contributions above it are positive.
- Antiderivative
- A function whose derivative equals the given integrand on the relevant interval; different antiderivatives differ by a constant there.
- Accumulation function
- A function defined by an integral with a variable endpoint, describing how a running total changes as that endpoint moves.
- Improper integral
- An integral requiring a limit because an endpoint is unbounded or another ordinary-integral condition fails at a boundary.
- Limit comparison
- A convergence test comparing the ratio of two nonnegative functions at infinity, with a positive finite ratio transferring convergence behaviour.
- Boundary term
- The endpoint evaluation of a product arising in integration by parts, which must be combined with the remaining integral using the correct sign.
Integration and Improper Integrals FAQ
Why can an integral be negative when area is positive?
A definite integral preserves the sign of the integrand, so contributions below the axis reduce the total. Geometric area counts all regions positively and therefore requires splitting or integrating the absolute value.
Should I always change bounds in a substitution?
You may either change the bounds to the new variable or substitute the primitive back before using the old bounds. Both approaches are valid; mixing new-variable expressions with old-variable bounds is the error.
How do I choose between substitution and parts?
Look for a chain-rule pattern first: an inner expression together with its derivative. For parts, choose a factor that simplifies when differentiated and another that can be integrated without making the remaining problem harder.
Does a comparison test calculate the integral?
Usually it establishes convergence or divergence without giving an exact value. A function can share the same tail behaviour as a simple reference while having a different integral because of its values over the finite portion.
Why must two infinite tails be checked separately?
Ordinary improper convergence requires both one-sided accumulated quantities to approach finite limits. Symmetric truncation can cancel divergent positive and negative contributions, producing a misleading finite-looking result that does not establish the required convergence.
Exam move
Pair every integration exercise with a different check. For a substitution, differentiate the returned primitive and inspect the transformed endpoints. For parts, differentiate the product expression and watch the cancellation. For a definite integral of a positive function, estimate a simple lower and upper bound before accepting the result. These checks catch distinct errors rather than repeating the same calculation.
Practise accumulation derivatives separately from integral evaluation so that the task verb controls your method. Then add improper examples in increasing difficulty: a directly integrable power tail, a limit-comparison problem, and a two-sided domain requiring separate tails. In each answer, write one clear sentence identifying what has been established.
Keep exact fractions and constants until the final line, and use a decimal only as a scale check. If a computation becomes complicated, return to the integrand's structure before reaching for another technique.
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