Area between curves — horizontal and vertical slices: Foundations

This page assumes you have seen nothing. We build every symbol the parent note leans on, in an order where each one only uses the ones before it. Follow from line one.

0. The coordinate plane — where everything lives

Before any curve, we need a place. Everything in this topic is drawn on a flat sheet with two number-lines crossing at a point called the origin.

Figure — Area between curves — horizontal and vertical slices

Why the topic needs it: every "curve" is just a rule that, for each spot you pick, tells you a height. Without the plane there is no "up," "down," "left," "right" — and this whole chapter is about which direction we sweep.

x

increases as you move which way?

To the right.

The point

(3, 2)

means what?

Go right 3, then up 2.

1. A function $y = f (x)$ — a machine that turns $x$ into a height

Why the topic needs it: the parent note constantly says "top curve $f (x)$ , bottom curve $g (x)$ ." Those are just two such machines. The letters $f$ and $g$ are only names — nothing more.

2. Two curves and the gap between them

Now put two machines on the same plane: $y = f (x)$ (call it the top) and $y = g (x)$ (the bottom), where at every $x$ in the region the top sits higher, written $f (x) \geq g (x)$ .

Why the topic needs it: $f (x) - g (x)$ is the star of the whole formula. It is the height of every vertical slice.

The vertical gap between top

f

and bottom

g

at position

x

f (x) - g (x)

, a non-negative length.

Why can't the gap come out negative in our setup?

Because we required

f (x) \geq g (x)

, so top minus bottom is

\geq 0

3. Chopping the interval: $[a, b]$ , $Δ x$ , $x_{i}^{*}$

We slice the region between two $x$ -values, $a$ (left edge) and $b$ (right edge). The stretch of $x$ from $a$ to $b$ is the interval $[a, b]$ .

Why the topic needs it: one thin rectangle has area (height) × (width) $= (f (x_{i}^{*}) - g (x_{i}^{*})) Δ x$ . Every rectangle is built from Section 2's gap times this thickness.

4. Adding the rectangles: the sum symbol $\sum$

Why the topic needs it: the total area of all the thin rectangles is $\sum_{i = 1}^{n} (f (x_{i}^{*}) - g (x_{i}^{*})) Δ x .$ This is exactly Section 2's gap $\times$ Section 3's thickness, added over every slice. This idea is developed fully in Definite Integral as Riemann Sum.

\sum_{i = 1}^{4} i

equals

1 + 2 + 3 + 4 = 10

5. The limit $n \to \infty$ and the integral $\int_{a}^{b}$

The rectangle-sum is only approximate — its stair-step top doesn't hug a curved boundary perfectly. Fix it by chopping infinitely finely.

Why the topic needs it: every area formula in the parent note is this one limit, just with different top/bottom (or right/left) curves plugged in.

The symbol

d x

is the limiting version of

the slice width

Δ x

as it shrinks to zero.

Why do we take

n \to \infty

To remove the staircase error and get the exact area under a curved boundary.

6. Reading the integral: the Fundamental Theorem shortcut

You never actually add infinitely many things by hand. Instead:

Why the topic needs it: this is the machine that turns Section 5's scary limit into arithmetic. Full story in Fundamental Theorem of Calculus.

Why must we check

\frac{d}{d x} F = f

before using

F

Because only an antiderivative of the integrand gives the correct area via

F (b) - F (a)

7. Switching viewpoints: $x = R (y)$ , $x = L (y)$ , and inverses

Sometimes it is cleaner to sweep bottom-to-top instead of left-to-right. Then we lay the rectangles sideways: width becomes $d y$ (thin height) and length becomes right curve − left curve.

Why the topic needs it: curves like $x = y^{2}$ hand you two $y$ -values for one $x$ , so vertical slicing needs splitting. Horizontal slicing reads $x = y^{2}$ directly — no split.

Right-minus-left formula for horizontal slices

A = \int_{c}^{d} (R (y) - L (y)) d y

To use horizontal slices on

y = x

, rewrite it as

x = y^{2}

8. Where do the limits come from? Solving for intersections

The edges $a, b$ (or $c, d$ ) are exactly where the two curves meet — the region is pinched shut there.

Why the topic needs it: you cannot even start the integral without the limits, and the limits are the intersection points.

Where do integration limits come from?

The points where the two curves intersect (solve

f = g

R = L

Prerequisite map

Equipment checklist

Cover the right side and test yourself. If any answer is shaky, reread that section above.

Read

f (x)

out loud and say what it is

f

x

" — the height/output of machine

f

at input

x

(not multiplication).

State the height of one vertical slice

top minus bottom,

f (x) - g (x)

, always

\geq 0

in our setup.

What is

Δ x

and what happens as

n

grows?

The width of one slice,

= \frac{b - a}{n}

; it shrinks toward

0

n \to \infty

Expand

\sum_{i = 1}^{3} i^{2}

1 + 4 + 9 = 14

Turn the rectangle sum into an integral

lim_{n \to \infty} \sum (f - g) Δ x = \int_{a}^{b} (f - g) d x

Compute

[x^{2} - \frac{x ^{3}}{3}]_{0}^{2}

(4 - \frac{8}{3}) - 0 = \frac{4}{3}

Write the horizontal-slice area formula

A = \int_{c}^{d} (R (y) - L (y)) d y

Invert

y = x

to an

x

-of-

y

form

x = y^{2}

Where do the integration limits come from?

The intersection points of the two curves.

Connections

Parent topic (Hinglish) — the note these foundations feed into.
Definite Integral as Riemann Sum — Sections 3–5 in full.
Fundamental Theorem of Calculus — Section 6, the antiderivative shortcut.
Inverse Functions — Section 7, rewriting $y = f (x)$ as $x = L (y)$ or $x = R (y)$ .
Solving Quadratic & Polynomial Equations — Section 8, finding the limits.
Volumes by Slicing & Disks — the same slice-measure-add pattern, one dimension higher.