Calculus, visually

Nine views of single-variable calculus. Each starts from a worked example in the book; pick another from Book examples… at the top of a tab, or type your own function. Functions use the graphing calculator's syntax: 21t^2, sqrt(t), sin(x)/x, e^(-x2), |t-1.5| (bars mean abs), x^3/4-x+2, with pi, e, ln, exp, abs, sign, cbrt and the trigonometric functions. Write the variable as x (or t). Number boxes accept expressions too: sqrt(pi), 2pi, 1e-4, and inf where an infinite limit makes sense.

Secant → tangent — a fixed point P and a point Q you drag along the curve. The secant, its slope (to six decimal places) and h update as you drag; every drag adds a row to the table of slopes from the right (h > 0) and from the left (h < 0). Fill adds h = ±1, ±0.5, ±0.1 … ±0.0001. The lower graph plots the secant slope against h. For |t-1.5| the two sides refuse to agree and there is no tangent.
Limits — the ε–δ game: the challenger sets a band of half-height ε around L; you drag a strip of half-width δ around a. The graph in the strip turns green inside the band and red where it escapes. Next round halves ε; Prover’s δ finds the largest δ that works, or shows that none does (a jump, once ε is below half the jump). One-sided limits tabulates f from the left and the right (or as x → ±∞) and says whether the sides agree, jump, or blow up.
Derivative from f — drag along f (or press Sweep): at each x the tangent’s slope is dropped as a dot underneath, and the dots trace out f′. Product rule draws f·g as a rectangle that grows by thin strips g·Δf and f·Δg and a corner Δf·Δg that vanishes as h → 0.
Riemann sums — left, right and midpoint rectangles, trapezoids and Simpson parabolas for 1–400 strips, with the sum, the exact integral (adaptive Gauss–Kronrod) and the error, a table of all five rules, and a log–log plot of error against n whose slopes (−1, −2, −2, −4) are the orders. Substitution shows strips of f(g(x))·g′(x) beside the matching strips of f(u): different shapes, equal areas. Improper integrates up to a cut-off and lets it run to ∞ (or to the point where f blows up), and says whether the area settles. Sum vs integral traps Σ f(k) between two integrals.
Fundamental theorem — paint the area under f(t) from a to x and watch A(x) = ∫ₐˣ f(t) dt underneath; the slope of A at x always equals f(x).
Area & volume — the area between two curves, split at their crossings; and the solid made by spinning a curve about the x-axis, by discs (or washers, with an inner radius).
Taylor — Tₙ about a for degree 0–20, a band of ± a tolerance around f with the interval where Tₙ stays inside it, the remainder |f − Tₙ| on a log scale, the Lagrange bound, and a table of coefficients and errors by degree.
Linearise & MVT — the tangent-line estimate L(x) = f(a) + f′(a)(x − a) with its error and the ½ f″(a)(x − a)² estimate; and the mean value theorem’s points c where the tangent is parallel to the chord.
Optimisation — the open-box designer (cut x from an a × b sheet, V(x) live, the optimum from the quadratic formula) and the cheapest can of a given volume when the ends cost k times the side.

On a phone, drag horizontally on a graph to move a point; vertical swipes scroll the page.

⛶ Full screen (or F) gives the active tab’s main plot the whole screen — the browser’s full screen where it is allowed, otherwise a full-window layer (iPhone) — redrawn at that size, with the controls beside it. Escape leaves.

Limits

Derivatives, Taylor coefficients and tangents are computed numerically (a Chebyshev fit around the point), to about 8–12 significant digits for smooth functions. A function that is not smooth at the point (a kink, a jump) has no Taylor polynomial, and the tool says so.
Tables of limits sample a few points, exactly as the book warns: try sin(pi/x) at 0. The ε–δ check samples about two thousand points in the strip plus points crowding in on a, so a pathological function can fool it.
The exact integral is a numerical one (adaptive Gauss–Kronrod, about 12 digits); errors below about 10⁻¹³ are rounding noise.
Improper integrals are judged from cut-offs up to 10⁷ (or distances down to 10⁻⁷); an integral that converges very slowly, such as ∫₁^∞ x^−1.1 dx, looks divergent at that range.

Related: the graphing calculator (plots, roots, derivatives and integrals of any function), symbolic algebra (exact derivatives, integrals, limits and Taylor series), the ODE solver (slope fields and Euler’s method), and algebra step by step.

Companion to chapters 16–19 of Mathematics, Reconnected: Limits, Derivatives, Applications of the derivative and Integrals. Each tab links to its section.

Linking to an example

Share link copies a URL that reopens the current tab exactly as it is set up. Links can also be written by hand: the tab is the part after #, and its settings are short query parameters, for example /tools/calculus/?f=sqrt(t)&a=1.5&h=0.1#secant or ?f=x2&a=0&b=2&n=8&method=mid#integrals. Parameters you leave out take the tab’s defaults; opening a link replaces that tab’s settings in this browser.

A + in a link means plus, so ?f=x^2+1 is x² + 1 (a space can be written %20, but is never needed). Other characters may be written plainly or percent-encoded (%5E for ^, %2F for /).

Parameter Meaning Example

Parameter	Meaning	Example
`#tab`	the tab: `secant`, `limit`, `derivative`, `integrals`, `ftc`, `area`, `taylor`, `linear`, `optimise`	`#taylor`
`mode`	the view inside a tab: `game` or `sides` (limit); `build` or `product` (derivative); `rules`, `subst`, `improper` or `sums` (integrals); `between` or `volume` (area); `linear` or `mvt` (linear); `box` or `can` (optimise)	`mode=sides`
`f`	the function (in x or t); in substitution the outer function f(u)	`f=21t^2`
`g`	a second function: g in the product rule, the second curve, the inner radius of a washer, or the inner function u = g(x)	`g=x^2`
`a`	the point (P, the limit point, Taylor’s centre, the linearisation point), or the left end of an interval; the box sheet’s length	`a=1.5`, `a=inf`
`b`	the right end of an interval (`inf` allowed for improper integrals); the box sheet’s width	`b=sqrt(pi)`
`h`	the secant’s step (Q = P + h), or the product rule’s nudge	`h=0.01`
`fill`	fill the secant table: `1` for h = ±1 … ±0.0001, or a list of h values	`fill=2,1.2,0.6,0.25`
`L`, `eps`, `delta`	the ε–δ game’s claimed limit, ε and δ	`L=7&eps=0.1&delta=0.05`
`n`	strips (Riemann), terms (sums), degree (Taylor)	`n=16`
`method`	the rule: `left`, `right`, `mid`, `trap` or `simpson`	`method=simpson`
`i`	the highlighted strip (substitution)	`i=3`
`cut`	the cut-off B (or distance c from the bad end) for an improper integral	`cut=1000`
`x`	the moving point: the endpoint (FTC), the evaluation point (Taylor, linearisation), the slice, the box’s cut	`x=0.5`
`lo`, `hi`	the plotted range (derivative, FTC, Taylor)	`lo=-pi&hi=2pi`
`tol`	the half-width of Taylor’s error band	`tol=0.001`
`r`, `vol`, `cost`	the can’s radius, volume, and the price of the ends relative to the side	`vol=500&cost=2`
`example`	a named example from the book (below); other parameters then adjust it	`example=ch16-car&h=0.01`

#tab

the tab: secant, limit, derivative, integrals, ftc, area, taylor, linear, optimise

#taylor

mode

the view inside a tab: game or sides (limit); build or product (derivative); rules, subst, improper or sums (integrals); between or volume (area); linear or mvt (linear); box or can (optimise)

mode=sides

f

the function (in x or t); in substitution the outer function f(u)

f=21t^2

g

a second function: g in the product rule, the second curve, the inner radius of a washer, or the inner function u = g(x)

g=x^2

a

the point (P, the limit point, Taylor’s centre, the linearisation point), or the left end of an interval; the box sheet’s length

a=1.5, a=inf

b

the right end of an interval (inf allowed for improper integrals); the box sheet’s width

b=sqrt(pi)

h

the secant’s step (Q = P + h), or the product rule’s nudge

h=0.01

fill

fill the secant table: 1 for h = ±1 … ±0.0001, or a list of h values

fill=2,1.2,0.6,0.25

L, eps, delta

the ε–δ game’s claimed limit, ε and δ

L=7&eps=0.1&delta=0.05

n

strips (Riemann), terms (sums), degree (Taylor)

n=16

method

the rule: left, right, mid, trap or simpson

method=simpson

i

the highlighted strip (substitution)

i=3

cut

the cut-off B (or distance c from the bad end) for an improper integral

cut=1000

x

the moving point: the endpoint (FTC), the evaluation point (Taylor, linearisation), the slice, the box’s cut

x=0.5

lo, hi

the plotted range (derivative, FTC, Taylor)

lo=-pi&hi=2pi

tol

the half-width of Taylor’s error band

tol=0.001

r, vol, cost

the can’s radius, volume, and the price of the ends relative to the side

vol=500&cost=2

example

a named example from the book (below); other parameters then adjust it

example=ch16-car&h=0.01

Examples from the book

Example Tab What it shows

Example	Tab	What it shows
`ch16-car`	Secant → tangent	d(t) = 21t² at t = 1.5: average speeds 84, 73.5, 65.1 … home in on 63 mph
`ch16-sqrt`	Secant → tangent	√t at t = 1.5: slopes approach 1/(2√1.5) ≈ 0.408248
`ch16-sin`	Secant → tangent	sin t at t = 1.5: slopes approach cos 1.5 ≈ 0.070737
`ch16-abs`	Secant → tangent	\|t − 1.5\|: right secants have slope 1, left ones −1 — no tangent
`ch16-hole`	Limits › One-sided limits	(x² − 4)/(x − 2) near 2: a hole, and the limit 4
`ch16-lying-table`	Limits › One-sided limits	sin(π/x): the table says 0, the graph says no limit
`ch16-sign`	Limits › One-sided limits	\|x\|/x: left limit −1, right limit 1 — a jump
`ch16-blowup`	Limits › One-sided limits	1/x² → ∞ at 0 (a vertical asymptote)
`ch16-one-over-x`	Limits › One-sided limits	1/x: −∞ from the left, +∞ from the right
`ch16-oscillation`	Limits › One-sided limits	sin(1/x): endless oscillation, no limit
`ch16-squeeze`	Limits › One-sided limits	x² sin(1/x): squeezed to 0
`ch16-sinx-over-x`	Limits › One-sided limits	sin x / x → 1
`ch16-rational-inf`	Limits › One-sided limits	(3x² − 5x + 1)/(2x² + 7) → 3/2 as x → ∞
`ch16-root-inf`	Limits › One-sided limits	√(x² + x) − x → 1/2 as x → ∞
`ch16-eps-linear`	Limits › ε–δ game	ε–δ for 2x + 1 at 3: δ = ε/2 always wins
`ch16-eps-quadratic`	Limits › ε–δ game	ε–δ for x² at 2: δ = min(1, ε/5) wins
`ch16-eps-sinc`	Limits › ε–δ game	ε–δ for sin x / x at 0 (f(0) undefined)
`ch16-eps-jump`	Limits › ε–δ game	ε–δ for a jump: no δ works once ε < 1/2
`ch17-cubic`	Secant → tangent	x³/4 − x + 2 at P = (1, 1.25): slopes 2.25, 1.01, 0.29, −0.046875 → −0.25
`ch17-abs`	Secant → tangent	\|x\| at P = (0, 0): slopes from the left and right refuse to agree
`ch17-base-2`	Secant → tangent	Slope of 2ˣ at 0: 0.71773, 0.69556, … → ln 2 ≈ 0.693
`ch17-base-e`	Secant → tangent	Slope of eˣ at 0 → exactly 1
`ch17-base-3`	Secant → tangent	Slope of 3ˣ at 0 → ln 3 ≈ 1.0986
`ch17-sin-slopes`	Derivative from f › Build f′ from slopes	Slopes of sin x trace out cos x
`ch17-cubic-slopes`	Derivative from f › Build f′ from slopes	Slopes of x³/4 − x + 2 trace the parabola 3x²/4 − 1
`ch17-product`	Derivative from f › Product rule	Product rule as a growing rectangle: (x² sin x)′ = 2x sin x + x² cos x
`ch18-mvt-sqrt`	Linearise & MVT › Mean value theorem	Mean value theorem for √x on [1, 4]: c = 2.25
`ch18-box`	Optimisation › Box designer	Box from a 20 cm sheet: best cut 10/3 cm, V ≈ 592.6 cm³
`ch18-box-rect`	Optimisation › Box designer	Box from a 20 × 32 sheet: the optimum moves
`ch18-can`	Optimisation › Can	500 cm³ can: r ≈ 4.301, h = 2r ≈ 8.603, area ≈ 348.7 cm²
`ch18-can-ends`	Optimisation › Can	Ends cost twice as much: h = 4r (r ≈ 3.41, h ≈ 13.66)
`ch18-sqrt-4.1`	Linearise & MVT › Linearisation	√4.1 ≈ 2.025 from the tangent at 4 (true 2.0248457…)
`ch18-small-angle`	Linearise & MVT › Linearisation	sin x ≈ x for small angles: sin 0.1 = 0.0998334
`ch18-cbrt`	Linearise & MVT › Linearisation	∛8.3 ≈ 2.025 (true 2.02469)
`ch18-ln`	Linearise & MVT › Linearisation	ln 1.05 ≈ 0.05 (true 0.04879)
`ch18-taylor-sin`	Taylor	sin x ≈ x − x³/6 + …: errors at 0.5 of 2.1e−2, 2.6e−4, 1.5e−6
`ch18-taylor-exp`	Taylor	e^0.1 ≈ 1 + 0.1 + 0.01/2 = 1.105
`ch18-taylor-ln`	Taylor	ln(1 + x): only converges for −1 < x ≤ 1, however many terms
`ch18-relativity`	Taylor	γ = (1 − u)^(−1/2) ≈ 1 + u/2: kinetic energy ½mv²
`ch19-riemann`	Riemann sums › Rules & errors	x² on [0, 2], n = 4: left 1.75, right 3.75, midpoint 2.625; exact 8/3
`ch19-riemann-decreasing`	Riemann sums › Rules & errors	A decreasing function: the left sum overestimates
`ch19-riemann-rise-fall`	Riemann sums › Rules & errors	sin x on [0, π]: neither rule is reliably under or over
`ch19-ln2`	Riemann sums › Rules & errors	ln 2 by hand: T₄ = 0.697024, S₄ = 0.693254
`ch19-sin-errors`	Riemann sums › Rules & errors	∫₀^π sin x = 2: trapezoid error ÷4, Simpson ÷16 per doubling
`ch19-exp-errors`	Riemann sums › Rules & errors	∫₀¹ eˣ: log–log slopes −1, −2, −2, −4
`ch19-simpson-cubic`	Riemann sums › Rules & errors	Simpson is exact for cubics: S₂ = 4, T₂ = 5
`ch19-midpoint-x2`	Riemann sums › Rules & errors	Midpoint, n = 2, for ∫₀⁴ x²: 20 against 64/3
`ch19-race-periodic`	Riemann sums › Rules & errors	Periodic over a full period: the trapezoid rule beats Simpson
`ch19-race-kink`	Riemann sums › Rules & errors	A kink: Simpson drops to order 2
`ch19-race-oscillating`	Riemann sums › Rules & errors	Oscillating: too few strips alias
`ch19-race-spike`	Riemann sums › Rules & errors	A spike: errors fall only once the strips resolve it
`ch19-ftc-dips`	Fundamental theorem	A(x) falls where f dips below the axis; slope of A = f(x)
`ch19-ftc-car`	Fundamental theorem	The car: distance A(3) = 120 miles
`ch19-ftc-sin`	Fundamental theorem	One arch of sin: area exactly 2
`ch19-ftc-gauss`	Fundamental theorem	∫₀ˣ e^(−t²) dt: no formula, but A′ = f
`ch19-substitution`	Riemann sums › Substitution	2x cos(x²) on [0, √π] ↔ cos u on [0, π]: same strip areas
`ch19-substitution-ex3`	Riemann sums › Substitution	∫₀¹ x(x² + 1)³ dx = ½∫₁² u³ du = 15/8
`ch19-area-between`	Area & volume › Between curves	Area between y = x and y = x²: 1/6
`ch19-area-2x`	Area & volume › Between curves	Area between 2x and x²: 4/3
`ch19-sphere`	Area & volume › Solid of revolution	Spin a semicircle: sphere volume 4/3 π r³
`ch19-cone`	Area & volume › Solid of revolution	Spin y = x on [0, 3]: a cone, 9π
`ch19-washer`	Area & volume › Solid of revolution	Washers: the region between x and x², spun: 2π/15
`ch19-horn`	Area & volume › Solid of revolution	Gabriel’s horn: volume → π as the horn grows
`ch19-improper-converges`	Riemann sums › Improper	∫₁^∞ 1/x² = 1: an infinite region of finite area
`ch19-improper-diverges`	Riemann sums › Improper	∫₁^∞ 1/x diverges like ln b
`ch19-improper-spike`	Riemann sums › Improper	∫₀¹ 1/√x = 2: a spike with finite area
`ch19-improper-exp`	Riemann sums › Improper	∫₀^∞ e^(−2x) = 1/2
`ch19-sum-loop`	Riemann sums › Sum vs integral	The nested loop: Σk = 36 for n = 8, between ∫₀⁸ x and ∫₁⁹ x
`ch19-harmonic`	Riemann sums › Sum vs integral	H₁₀₀₀ ≈ 7.485, between ln 1001 and 1 + ln 1000
`ch19-sum-squares`	Riemann sums › Sum vs integral	Σk² grows like n³/3
`ch19-sum-sqrt`	Riemann sums › Sum vs integral	Σ√k grows like ⅔ n^(3/2)
`ch19-log-factorial`	Riemann sums › Sum vs integral	ln n! ≈ n ln n − n: the sorting lower bound

ch16-car

Secant → tangent

d(t) = 21t² at t = 1.5: average speeds 84, 73.5, 65.1 … home in on 63 mph

ch16-sqrt

Secant → tangent

√t at t = 1.5: slopes approach 1/(2√1.5) ≈ 0.408248

ch16-sin

Secant → tangent

sin t at t = 1.5: slopes approach cos 1.5 ≈ 0.070737

ch16-abs

Secant → tangent

|t − 1.5|: right secants have slope 1, left ones −1 — no tangent

ch16-hole

Limits › One-sided limits

(x² − 4)/(x − 2) near 2: a hole, and the limit 4

ch16-lying-table

Limits › One-sided limits

sin(π/x): the table says 0, the graph says no limit

ch16-sign

Limits › One-sided limits

|x|/x: left limit −1, right limit 1 — a jump

ch16-blowup

Limits › One-sided limits

1/x² → ∞ at 0 (a vertical asymptote)

ch16-one-over-x

Limits › One-sided limits

1/x: −∞ from the left, +∞ from the right

ch16-oscillation

Limits › One-sided limits

sin(1/x): endless oscillation, no limit

ch16-squeeze

Limits › One-sided limits

x² sin(1/x): squeezed to 0

ch16-sinx-over-x

Limits › One-sided limits

sin x / x → 1

ch16-rational-inf

Limits › One-sided limits

(3x² − 5x + 1)/(2x² + 7) → 3/2 as x → ∞

ch16-root-inf

Limits › One-sided limits

√(x² + x) − x → 1/2 as x → ∞

ch16-eps-linear

Limits › ε–δ game

ε–δ for 2x + 1 at 3: δ = ε/2 always wins

ch16-eps-quadratic

Limits › ε–δ game

ε–δ for x² at 2: δ = min(1, ε/5) wins

ch16-eps-sinc

Limits › ε–δ game

ε–δ for sin x / x at 0 (f(0) undefined)

ch16-eps-jump

Limits › ε–δ game

ε–δ for a jump: no δ works once ε < 1/2

ch17-cubic

Secant → tangent

x³/4 − x + 2 at P = (1, 1.25): slopes 2.25, 1.01, 0.29, −0.046875 → −0.25

ch17-abs

Secant → tangent

|x| at P = (0, 0): slopes from the left and right refuse to agree

ch17-base-2

Secant → tangent

Slope of 2ˣ at 0: 0.71773, 0.69556, … → ln 2 ≈ 0.693

ch17-base-e

Secant → tangent

Slope of eˣ at 0 → exactly 1

ch17-base-3

Secant → tangent

Slope of 3ˣ at 0 → ln 3 ≈ 1.0986

ch17-sin-slopes

Derivative from f › Build f′ from slopes

Slopes of sin x trace out cos x

ch17-cubic-slopes

Derivative from f › Build f′ from slopes

Slopes of x³/4 − x + 2 trace the parabola 3x²/4 − 1

ch17-product

Derivative from f › Product rule

Product rule as a growing rectangle: (x² sin x)′ = 2x sin x + x² cos x

ch18-mvt-sqrt

Linearise & MVT › Mean value theorem

Mean value theorem for √x on [1, 4]: c = 2.25

ch18-box

Optimisation › Box designer

Box from a 20 cm sheet: best cut 10/3 cm, V ≈ 592.6 cm³

ch18-box-rect

Optimisation › Box designer

Box from a 20 × 32 sheet: the optimum moves

ch18-can

Optimisation › Can

500 cm³ can: r ≈ 4.301, h = 2r ≈ 8.603, area ≈ 348.7 cm²

ch18-can-ends

Optimisation › Can

Ends cost twice as much: h = 4r (r ≈ 3.41, h ≈ 13.66)

ch18-sqrt-4.1

Linearise & MVT › Linearisation

√4.1 ≈ 2.025 from the tangent at 4 (true 2.0248457…)

ch18-small-angle

Linearise & MVT › Linearisation

sin x ≈ x for small angles: sin 0.1 = 0.0998334

ch18-cbrt

Linearise & MVT › Linearisation

∛8.3 ≈ 2.025 (true 2.02469)

ch18-ln

Linearise & MVT › Linearisation

ln 1.05 ≈ 0.05 (true 0.04879)

ch18-taylor-sin

Taylor

sin x ≈ x − x³/6 + …: errors at 0.5 of 2.1e−2, 2.6e−4, 1.5e−6

ch18-taylor-exp

Taylor

e^0.1 ≈ 1 + 0.1 + 0.01/2 = 1.105

ch18-taylor-ln

Taylor

ln(1 + x): only converges for −1 < x ≤ 1, however many terms

ch18-relativity

Taylor

γ = (1 − u)^(−1/2) ≈ 1 + u/2: kinetic energy ½mv²

ch19-riemann

Riemann sums › Rules & errors

x² on [0, 2], n = 4: left 1.75, right 3.75, midpoint 2.625; exact 8/3

ch19-riemann-decreasing

Riemann sums › Rules & errors

A decreasing function: the left sum overestimates

ch19-riemann-rise-fall

Riemann sums › Rules & errors

sin x on [0, π]: neither rule is reliably under or over

ch19-ln2

Riemann sums › Rules & errors

ln 2 by hand: T₄ = 0.697024, S₄ = 0.693254

ch19-sin-errors

Riemann sums › Rules & errors

∫₀^π sin x = 2: trapezoid error ÷4, Simpson ÷16 per doubling

ch19-exp-errors

Riemann sums › Rules & errors

∫₀¹ eˣ: log–log slopes −1, −2, −2, −4

ch19-simpson-cubic

Riemann sums › Rules & errors

Simpson is exact for cubics: S₂ = 4, T₂ = 5

ch19-midpoint-x2

Riemann sums › Rules & errors

Midpoint, n = 2, for ∫₀⁴ x²: 20 against 64/3

ch19-race-periodic

Riemann sums › Rules & errors

Periodic over a full period: the trapezoid rule beats Simpson

ch19-race-kink

Riemann sums › Rules & errors

A kink: Simpson drops to order 2

ch19-race-oscillating

Riemann sums › Rules & errors

Oscillating: too few strips alias

ch19-race-spike

Riemann sums › Rules & errors

A spike: errors fall only once the strips resolve it

ch19-ftc-dips

Fundamental theorem

A(x) falls where f dips below the axis; slope of A = f(x)

ch19-ftc-car

Fundamental theorem

The car: distance A(3) = 120 miles

ch19-ftc-sin

Fundamental theorem

One arch of sin: area exactly 2

ch19-ftc-gauss

Fundamental theorem

∫₀ˣ e^(−t²) dt: no formula, but A′ = f

ch19-substitution

Riemann sums › Substitution

2x cos(x²) on [0, √π] ↔ cos u on [0, π]: same strip areas

ch19-substitution-ex3

Riemann sums › Substitution

∫₀¹ x(x² + 1)³ dx = ½∫₁² u³ du = 15/8

ch19-area-between

Area & volume › Between curves

Area between y = x and y = x²: 1/6

ch19-area-2x

Area & volume › Between curves

Area between 2x and x²: 4/3

ch19-sphere

Area & volume › Solid of revolution

Spin a semicircle: sphere volume 4/3 π r³

ch19-cone

Area & volume › Solid of revolution

Spin y = x on [0, 3]: a cone, 9π

ch19-washer

Area & volume › Solid of revolution

Washers: the region between x and x², spun: 2π/15

ch19-horn

Area & volume › Solid of revolution

Gabriel’s horn: volume → π as the horn grows

ch19-improper-converges

Riemann sums › Improper

∫₁^∞ 1/x² = 1: an infinite region of finite area

ch19-improper-diverges

Riemann sums › Improper

∫₁^∞ 1/x diverges like ln b

ch19-improper-spike

Riemann sums › Improper

∫₀¹ 1/√x = 2: a spike with finite area

ch19-improper-exp

Riemann sums › Improper

∫₀^∞ e^(−2x) = 1/2

ch19-sum-loop