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2. Calculus — Integration

Riemann integration

Riemann integration rigorously formalizes the concept of definite integrals as limits of lower and upper sums associated with partitions of a closed interval.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade
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Definition and statement

Riemann Integrability Criterion (Darbuox's Theorem)
A bounded function $f: [a, b] \to \mathbb{R}$ is Riemann integrable on $[a, b]$ (written as $f \in \mathcal{R}[a, b]$) if and only if for every $\epsilon > 0$, there exists a partition $P$ of $[a, b]$ such that the upper Darboux sum $U(P, f)$ and lower Darboux sum $L(P, f)$ satisfy $U(P, f) - L(P, f) < \epsilon$. Equivalently, $\underline{\int_a^b} f(x)\,dx = \overline{\int_a^b} f(x)\,dx = \int_a^b f(x)\,dx$.

📐 Geometric Interpretation

Geometrically, Riemann integration measures the net area bounded by the curve $y = f(x)$ and the $x$-axis by sandwiching the area between inscribed step polygons (lower sums) and circumscribed step polygons (upper sums). As the mesh size $\mu(P) \to 0$, the gap between these approximations vanishes.

⚙️ Algebraic & Structural Meaning

It establishes a bridge between set partitions, supremum/infimum operations in ordered fields, and limit processes, laying the rigorous foundation for calculus beyond elementary antiderivatives.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition Domain / Interval Necessity What Fails If Condition Dropped
Boundedness on $[a, b]$ [a, b] (Closed interval) Strictly Mandatory / Necessary If $f$ is unbounded (e.g., $f(x) = 1/\sqrt{x}$ near $0$), standard Riemann sums diverge to infinity, requiring improper Riemann integrals.
Continuity on $[a, b]$ [a, b] (Closed interval) Sufficient but not Necessary Discontinuous functions may still be Riemann integrable provided their set of discontinuities has measure zero (Lebesgue's Criterion).
Partition Norm $\mu(P) \to 0$ Partition refinement Essential for convergence If the mesh size does not approach zero, Riemann sums fail to converge to the true definite integral.

Useful results and shortcuts

Use these identities and results when their conditions apply:

01

Limit of Sums to Definite Integral Conversion

$$\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^{an} f\left(\frac{r}{n}\right) = \int_0^a f(x)\,dx$$

Rule: Replace $\frac{1}{n}$ with $dx$, $\frac{r}{n}$ with $x$, and $\lim \frac{1}{n} \sum$ with $\int$. Set lower limit to $x = \lim_{n\to\infty} \frac{1}{n} = 0$ and upper limit to $\lim_{n\to\infty} \frac{an}{n} = a$.

Application: Frequently asked in UP PGT/TGT and KVS PGT to evaluate complex series limits in seconds.
02

General Form with Lower and Upper Limits

$$\lim_{n \to \infty} \frac{1}{n} \sum_{r=an}^{bn} f\left(\frac{r}{n}\right) = \int_a^b f(x)\,dx$$

Rule: Substitute $\frac{r}{n} \to x$, $\frac{1}{n} \to dx$, lower limit $a = \lim \frac{an}{n}$, upper limit $b = \lim \frac{bn}{n}$.

Application: Direct PYQ pattern in DSSSB PGT Mathematics.
03

Power Sum Series Limit

$$\lim_{n \to \infty} \frac{1}{n^{k+1}} \sum_{r=1}^{n} r^k = \int_0^1 x^k dx = \frac{1}{k+1}$$

Rule: Factor out $1/n$ and $1/n^k$, convert the sum to $\int_0^1 x^k dx$, and integrate immediately.

04

Even and Odd Function Integration on Symmetric Intervals

$$\int_{-a}^{a} f(x)\,dx = \begin{cases} 2 \int_0^a f(x)\,dx, & \text{if } f(-x) = f(x) \\ 0, & \text{if } f(-x) = -f(x) \end{cases}$$

Rule: Check parity immediately. If odd, mark 0 without computing.

05

King's Property (Definite Integral Reversal)

$$\int_a^b f(x)\,dx = \int_a^b f(a + b - x)\,dx$$

Rule: Add the original integral and transformed integral to simplify logarithmic and trigonometric denominators.

06

Leibniz Rule for Differentiation under Integral Sign

$$\frac{d}{dx} \left[ \int_{\phi(x)}^{\psi(x)} f(t)\,dt \right] = f(\psi(x)) \cdot \psi'(x) - f(\phi(x)) \cdot \phi'(x)$$

Rule: Differentiate upper limit and substitute into integrand times derivative of upper limit, minus lower limit substitution times its derivative.

Application: Use this result in the stated type of problem.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

Unbounded Functions and Riemann Integrability

Common Pitfall: Assuming every continuous function on $[a, b]$ is Riemann integrable, or forgetting that unbounded functions fail standard Riemann integrability.

Counterexample: $f(x) = \frac{1}{x}$ on $[0, 1]$

Why it fails: The function is unbounded near $0$, hence not Riemann integrable in the proper sense.

🛡️ Exam Defense: Always verify that the domain interval is closed and bounded and the function is bounded before applying Riemann sum formulas.
⚠️ COMMON ERROR #02

Incorrect Limits in Sum-to-Integral Conversion

Common Pitfall: Blindly taking limits from $0$ to $1$ when the summation index runs from $r = an$ to $bn$.

Counterexample: $$\lim_{n \to \infty} \sum_{r=n}^{2n} \frac{1}{n} f\left(\frac{r}{n}\right)$$

Why it fails: Lower limit is $1$ and upper limit is $2$, not $0$ and $1$.

🛡️ Exam Defense: Always evaluate lower limit as $\lim_{n\to\infty} \frac{\text{lower index}}{n}$ and upper limit as $\lim_{n\to\infty} \frac{\text{upper index}}{n}$.

Comparison

Concept / TheoremHypotheses / AssumptionsKey Formula / OutcomeGeometric MeaningExam Identification Clue
Riemann Sum LimitBounded function, Mesh $\to 0$$$\lim \sum f(x_r) \Delta x_r$$Area of approximating rectanglesSum with $1/n$ and $r/n$ terms
Darboux IntegrabilityBounded function on $[a, b]$$U(P, f) = L(P, f)$Supremum of lower sums equals infimum of upper sumsQuestions testing oscillation and partitions
Fundamental Theorem (Part 1)Continuity on $[a, b]$$$\frac{d}{dx} \int_a^x f(t)dt = f(x)$$Rate of change of accumulated area equals heightDerivative of an integral expression
Fundamental Theorem (Part 2)Integrability + Antiderivative $F$$$\int_a^b f(x)dx = F(b) - F(a)$$Net accumulated area under curveStandard definite integral evaluation

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

Evaluate the limit: $$\lim_{n \to \infty} \left( \frac{1}{n+1} + \frac{1}{n+2} + \dots + \frac{1}{2n} \right)$$

(A) $\log 2$(B) $\log 3$(C) $\frac{\pi}{4}$(D) $1$
✓ Correct Answer: (A)
📚 Method A: Formal Textbook Derivation
Rewrite as $\sum_{r=1}^{n} \frac{1}{n+r} = \frac{1}{n} \sum_{r=1}^{n} \frac{1}{1 + r/n}$. As $n \to \infty$, this converts to $\int_0^1 \frac{1}{1+x} dx = [\log(1+x)]_0^1 = \log 2$.
⚡ Method B: 25-Second Exam Shortcut
Identify lower limit $1$ and upper limit $2$ inside $\frac{1}{n} \sum_{r=n}^{2n} \frac{1}{n}$ or convert directly to $\int_0^1 \frac{dx}{1+x} = \log 2$.
💡 Tip: Recognize the harmonic progression sum format immediately.
CANONICAL PROBLEM 02 Dual-Method Breakdown

Evaluate: $$\lim_{n \to \infty} \frac{1^4 + 2^4 + 3^4 + \dots + n^4}{n^5}$$

(A) $\frac{1}{4}$(B) $\frac{1}{5}$(C) $\frac{1}{30}$(D) $\frac{1}{15}$
✓ Correct Answer: (B)
📚 Method A: Formal Textbook Derivation
Express as $\lim_{n \to \infty} \frac{1}{n} \sum_{r=1}^n \left(\frac{r}{n}\right)^4$. By definite integral limit definition, this equals $\int_0^1 x^4 dx = \left[ \frac{x^5}{5} \right]_0^1 = \frac{1}{5}$.
⚡ Method B: 25-Second Exam Shortcut
Power sum limit formula: for $\sum r^k / n^{k+1}$, answer is always $\frac{1}{k+1} = \frac{1}{5}$.
💡 Tip: Direct 5-second template application for polynomial sums.

Practice questions

Choose an answer to check it and see the solution:

Q1
MCQ 1: The value of $$\lim_{n \to \infty} \sum_{r=1}^{n} \frac{n}{n^2 + r^2}$$ is
A $\frac{\pi}{4}$
B $\frac{\pi}{2}$
C $\log 2$
D $\pi$
Derivation & Analysis:

Rewrite as $\lim_{n\to\infty} \frac{1}{n} \sum_{r=1}^n \frac{1}{1 + (r/n)^2} = \int_0^1 \frac{dx}{1+x^2} = [\tan^{-1}x]_0^1 = \frac{\pi}{4}$.

Q2
MCQ 2: A bounded function $f$ on $[a, b]$ is Riemann integrable if and only if
A $\underline{\int} f = \overline{\int} f$
B $f$ is continuous everywhere
C $f$ is differentiable
D $U(P, f) \neq L(P, f)$
Derivation & Analysis:

By definition, Riemann integrability requires lower integral equal to upper integral.

Q3
MCQ 3: The value of $$\lim_{n \to \infty} \left( \frac{1}{n} \sum_{r=1}^{2n} \frac{r}{\sqrt{n^2 + r^2}} \right)$$ is
A $\sqrt{5} - 1$
B $\sqrt{2} - 1$
C $2-\sqrt{2}$
D $\sqrt{3} - 1$
Derivation & Analysis:

Convert to $\int_0^2 \frac{x}{\sqrt{1+x^2}} dx = [\sqrt{1+x^2}]_0^2 = \sqrt{5} - 1$. Wait, limits are from $0$ to $2$, so $\sqrt{1+4} - \sqrt{1} = \sqrt{5}-1$.

Q4
MCQ 4: If $f(x)$ is continuous on $[a, b]$, then $\int_a^b f(x)\,dx$ exists because
A Continuous functions are bounded and uniformly continuous on closed intervals
B Every function has an antiderivative
C Derivatives are always integrable
D Partition sums are constant
Derivation & Analysis:

Heine-Cantor theorem guarantees uniform continuity on compact sets $[a, b]$, ensuring Riemann integrability.

Q5
MCQ 5: The value of $$\int_0^{\pi/2} \log(\tan x)\,dx$$ is
A $1$
B $0$
C $\pi/2$
D $\log 2$
Derivation & Analysis:

Using King's property, $\int_0^{\pi/2} \log(\tan(\pi/2 - x))dx = \int_0^{\pi/2} \log(\cot x)dx = - \int_0^{\pi/2} \log(\tan x)dx$. Summing gives $2I = 0 \implies I = 0$.

Q6
MCQ 6: The limit $$\lim_{n \to \infty} \frac{1}{n} \left( e^{1/n} + e^{2/n} + \dots + e^{n/n} \right)$$ equals
A $e$
B $e - 1$
C $\log e$
D $1$
Derivation & Analysis:

Convert to $\int_0^1 e^x dx = [e^x]_0^1 = e - 1$.

Q7
MCQ 7: Which of the following functions is NOT Riemann integrable on $[0, 1]$?
A $f(x) = x^2$
B $f(x) = \sin x$
C $f(x) = \text{Dirichlet function}$
D $f(x) = |x|$
Derivation & Analysis:

The Dirichlet function is discontinuous everywhere on $[0, 1]$, and its lower and upper Darboux sums are $0$ and $1$ respectively, so it is not Riemann integrable.

Q8
MCQ 8: If $F(x) = \int_{0}^{x^2} \cos(t^2)\,dt$, then $F'(x)$ is
A $2x \cos(x^4)$
B $\cos(x^2)$
C $2x \cos(x^2)$
D $\cos(x^4)$
Derivation & Analysis:

Applying Leibniz Rule: $\cos((x^2)^2) \cdot \frac{d}{dx}(x^2) = 2x \cos(x^4)$.

Q9
MCQ 9: The value of $$\lim_{n \to \infty} \left( \frac{n!}{n^n} \right)^{1/n}$$ can be evaluated using integration as
A $1/e$
B $e$
C $1$
D $0$
Derivation & Analysis:

Taking log and applying Stolz-Cesaro or Riemann sum yields $\int_0^1 \log x dx = -1$, hence $e^{-1} = 1/e$.

Q10
MCQ 10: If $f:[a, b] \to \mathbb{R}$ is monotonic, then $f$ is
A Always Riemann integrable
B Never Riemann integrable
C Differentiable everywhere
D Continuous everywhere
Derivation & Analysis:

A monotonic function on a closed bounded interval has at most countably many discontinuities of the first kind, hence by Lebesgue's criterion it is Riemann integrable.

Common questions

Yes! Any function continuous on a closed and bounded interval $[a, b]$ is uniformly continuous and consequently Riemann integrable on $[a, b].

Candidates often misidentify the lower and upper limits of integration by failing to evaluate $\lim_{n\to\infty} \frac{\text{index}}{n}$ correctly, leading to incorrect integration bounds.

Memorize the template: replace $1/n$ with $dx$, $r/n$ with $x$, $\sum$ with $\int$, and determine limits by inspecting the starting and ending values of $r$ divided by $n$.

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