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What is Calculus II

an Integral?

The accumulation of infinitely many infinitesimal quantities — area, volume, total change, and the reverse of differentiation.

Simple Definition

Imagine you're driving and your speedometer shows your speed at every moment. The integral of your speed over a time period gives you the total distance travelled. It's the accumulation of all those tiny little speed × time pieces added together.

Geometrically, the integral of a function over an interval is the area between the curve and the x-axis. If you sliced that area into infinitely many infinitely thin rectangles and added them all up, that's what an integral computes.

There are two kinds: an indefinite integral gives you a family of functions (the antiderivative), while a definite integral gives you a specific number — the accumulated total between two bounds.

Intuition

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Slicing into rectangles

Divide the area under a curve into $n$ thin vertical rectangles. Each rectangle has width $\Delta x$ and height $f(x_i)$, so its area is $f(x_i)\,\Delta x$. Add them all up: $\sum f(x_i)\,\Delta x$. Now let $n \to \infty$ so the rectangles become infinitely thin — the sum becomes an integral: $\int f(x)\,dx$. This is the Riemann sum definition.

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Integration is the reverse of differentiation

If the derivative asks "how fast is this changing?", the integral asks "given how fast it's changing, what's the total accumulated change?" They are inverse operations — which is exactly what the Fundamental Theorem of Calculus states. That's why we call the indefinite integral an antiderivative.

Visual — Riemann Sum → Integral

Rectangles: 8

Drag the slider — as n → ∞, the Riemann sum converges to the exact integral

Definite vs Indefinite

Indefinite Integral

$$\int f(x)\,dx = F(x) + C$$

Returns a function — the antiderivative of $f$. The constant $C$ represents all possible vertical shifts. No bounds, no specific number.

Definite Integral

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

Returns a number — the net signed area between $f$ and the x-axis from $a$ to $b$. Uses the Fundamental Theorem of Calculus.

Formal Definition

The definite integral is defined as the limit of Riemann sums:

Riemann Sum Definition

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

Here $x_i^*$ is a sample point in the $i$-th subinterval. As $n \to \infty$ the rectangles become infinitely thin and the sum converges to the exact area.

Key Integration Rules

Power Rule	$\displaystyle\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$
Constant	$\displaystyle\int c\,dx = cx + C$
Exponential	$\displaystyle\int e^x\,dx = e^x + C$
Natural Log	$\displaystyle\int \frac{1}{x}\,dx = \ln\|x\| + C$
Sine	$\displaystyle\int \sin x\,dx = -\cos x + C$
Cosine	$\displaystyle\int \cos x\,dx = \sin x + C$
Sum Rule	$\displaystyle\int [f+g]\,dx = \int f\,dx + \int g\,dx$
Constant Multiple	$\displaystyle\int c\cdot f\,dx = c\int f\,dx$

Quick Examples

Example 1 — Indefinite

$\displaystyle\int x^3\,dx$

Power rule: $\dfrac{x^4}{4} + C$

Example 2 — Definite

$\displaystyle\int_0^2 x^2\,dx$

$\Big[\dfrac{x^3}{3}\Big]_0^2 = \dfrac{8}{3} - 0 = \dfrac{8}{3}$

Example 3 — U-Sub

$\displaystyle\int 2x\,e^{x^2}\,dx$

Let $u=x^2$, $du=2x\,dx$:
$\displaystyle\int e^u\,du = e^{x^2}+C$

Example 4 — Area

$\displaystyle\int_0^\pi \sin x\,dx$

$\Big[-\cos x\Big]_0^\pi = -(-1)-(-1) = 2$
Area = 2 square units

Common Mistakes

✗ Forgetting the constant C

Every indefinite integral must include $+C$. Without it, you're claiming one specific antiderivative — but there are infinitely many, all differing by a constant. Losing $C$ is a guaranteed point deduction.

✗ Applying the power rule to 1/x

$\displaystyle\int \frac{1}{x}\,dx \neq \frac{x^0}{0}$. That's division by zero. The correct answer is $\ln|x| + C$. The power rule $\frac{x^{n+1}}{n+1}$ breaks down at $n = -1$.

✗ Confusing area with signed area

A definite integral gives signed area — regions below the x-axis contribute negatively. If you want total (unsigned) area, you must integrate $|f(x)|$, splitting at every zero crossing.

Related Concepts

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Power Rule	\(\displaystyle\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)\)
Constant	\(\displaystyle\int c\,dx = cx + C\)
Exponential	\(\displaystyle\int e^x\,dx = e^x + C\)
Natural Log	\(\displaystyle\int \frac{1}{x}\,dx = \ln\|x\| + C\)
Sine	\(\displaystyle\int \sin x\,dx = -\cos x + C\)
Cosine	\(\displaystyle\int \cos x\,dx = \sin x + C\)
Sum Rule	\(\displaystyle\int [f+g]\,dx = \int f\,dx + \int g\,dx\)
Constant Multiple	\(\displaystyle\int c\cdot f\,dx = c\int f\,dx\)