Definite IntegralsJEE Main

Area under a curve via Riemann sums — interactive Mathematics simulation for IIT-JEE.

Concept

A definite integral $\int_a^b f(x)\,dx$ is the signed area between the curve and the x-axis. A Riemann sum approximates it with rectangles; as their number $n\to\infty$ (width $\to 0$ ) the sum converges to the exact area.

Key formula

\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^{n} f(x_i^*)\,\Delta x,\quad \Delta x=\frac{b-a}{n}

Derivation

Partition $[a,b]$ into $n$ strips of width $\Delta x=\frac{b-a}{n}$ . Each strip's area is approximated by a rectangle of height $f(x_i^*)$ (here the midpoint).

Summing gives $S_n=\sum_i f(x_i^*)\Delta x$ . By the Fundamental Theorem of Calculus, the limit equals $F(b)-F(a)$ where $F'=f$ .

For $f(x)=kx^2+1$ : $F(x)=\frac{kx^3}{3}+x$ , so $\int_a^b f = \big[\tfrac{kx^3}{3}+x\big]_a^b$ .

Scenarios to explore

Definite Integrals — Area under a curve via Riemann sums.
Few rectangles: a rough estimate — Only three strips: the sum drifts away from the exact area.
Many rectangles: the limit — Forty strips are indistinguishable from the exact area.

Real-world applications

Area, volume (solids of revolution) and arc length.
Work done by a variable force; centre of mass.
Average value of a function over an interval.

JEE exam tips

Midpoint sums converge faster than left/right sums for smooth curves.
Use symmetry: $\int_{-a}^{a}$ of an odd function is 0, even doubles the half.

Common mistakes

Forgetting Δx in the sum — the rectangles need a width.
Ignoring sign: area below the axis contributes negatively.
Swapping limits without the sign change: $\int_a^b=-\int_b^a$ .

Exam traps to avoid

The Riemann sum is an approximation; only the limit equals the integral.
Discontinuities or sign changes inside [a,b] need care — split the interval.