CALCULUS · ACCUMULATION

Integrals

Integrals measure accumulation. They connect infinitely many tiny contributions to total change, signed area, average value and antiderivatives — and the Fundamental Theorem of Calculus ties all of this directly to derivatives.

\[\int_a^b f(x)\,dx=F(b)-F(a)\]when \(F'(x)=f(x)\)

1. What an integral means

The definite integral \(\int_a^b f(x)\,dx\) is the net accumulation of \(f\) from \(a\) to \(b\). Geometrically, it is signed area: regions above the x-axis contribute positively and regions below contribute negatively.

Accumulation

Add infinitely many tiny contributions \(f(x)\,dx\).

Signed area

Above the axis is positive; below is negative.

Total change

If \(r(t)\) is a rate, \(\int r(t)dt\) gives accumulated change.

The indefinite integral describes all antiderivatives:

\[\int f(x)\,dx=F(x)+C,\qquad F'(x)=f(x).\]
Do not confuse the two. A definite integral is a number. An indefinite integral is a family of functions differing by a constant.

2. From rectangles to a Riemann integral

Partition \([a,b]\) into \(n\) subintervals of width \(\Delta x=(b-a)/n\). Choose a sample point \(x_i^*\) in each interval. The sum

\[S_n=\sum_{i=1}^{n}f(x_i^*)\,\Delta x\]

approximates the accumulated area. If these sums approach a unique limit as the partition becomes arbitrarily fine, then

\[\int_a^b f(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^n f(x_i^*)\Delta x.\]

Every continuous function on a closed interval is Riemann integrable. Many discontinuous functions are integrable too, but continuity is a powerful sufficient condition.

3. Fundamental Theorem of Calculus

The theorem explains why differentiation and integration are inverse processes.

Part I — accumulation creates an antiderivative

\[A(x)=\int_a^x f(t)\,dt\quad\Longrightarrow\quad A'(x)=f(x)\]

for continuous \(f\). The instantaneous rate of accumulated area is exactly the current function value.

Part II — evaluate definite integrals with antiderivatives

\[\int_a^b f(x)\,dx=F(b)-F(a),\qquad F'=f.\]

This replaces an infinite limiting sum by two function evaluations.

4. Core integration rules

\[\int x^n dx=\frac{x^{n+1}}{n+1}+C\quad(n\ne-1)\]
\[\int \frac1x dx=\ln|x|+C,\qquad \int e^x dx=e^x+C\]
\[\int \sin x\,dx=-\cos x+C,\qquad \int \cos x\,dx=\sin x+C\]

Linearity lets constants and sums pass through the integral:

\[\int(af+bg)dx=a\int fdx+b\int gdx.\]

Substitution

Substitution reverses the chain rule. If \(u=g(x)\), then \(du=g'(x)dx\):

\[\int f(g(x))g'(x)\,dx=\int f(u)\,du.\]

Integration by parts

This reverses the product rule:

\[\int u\,dv=uv-\int v\,du.\]

5. Worked examples

A — polynomial definite integral

Evaluate \(\int_0^2(3x^2-2x+1)dx\).

\[F(x)=x^3-x^2+x\Rightarrow F(2)-F(0)=8-4+2=6.\]

B — substitution

Evaluate \(\int 2x\cos(x^2)dx\). Set \(u=x^2\), so \(du=2x dx\).

\[\int\cos u\,du=\sin u+C=\sin(x^2)+C.\]

C — area versus signed integral

For \(f(x)=x\) on \([-1,1]\), symmetry gives \(\int_{-1}^{1}x\,dx=0\), but the geometric area between the graph and x-axis is \(1\). To compute total geometric area, integrate \(|f(x)|\) or split where the sign changes.

D — average value

The average value of continuous \(f\) on \([a,b]\) is

\[f_{avg}=\frac1{b-a}\int_a^b f(x)dx.\]

6. Interactive polynomial integral & Riemann tool

Explore \(f(x)=Ax^3+Bx^2+Cx+D\). The tool computes the exact definite integral and visualizes midpoint rectangles.

7. Common mistakes

  • Forgetting the constant \(+C\) in an indefinite integral.
  • Treating a definite integral as ordinary geometric area when the function changes sign.
  • Using the power rule with \(n=-1\); \(\int x^{-1}dx=\ln|x|+C\).
  • Changing variables in substitution but forgetting to change the differential or bounds.
  • Assuming every antiderivative has an elementary closed form; \(\int e^{-x^2}dx\) is a classic counterexample.

8. Practice

  1. Find \(\int(4x^3-6x+2)dx\).
  2. Evaluate \(\int_1^3 2x\,dx\).
  3. Evaluate \(\int_0^1 3x^2e^{x^3}dx\) by substitution.
  4. Find the average value of \(f(x)=x^2\) on \([0,3]\).
Solutions

1) \(x^4-3x^2+2x+C\). 2) \([x^2]_1^3=8\). 3) \(e-1\). 4) \(3\).