Calculus-Master
22

Day 22 · Review + Quiz

Phase 3 Review

Every formula from Day 16–21 in one place. Skim it, then take the quiz below when you're ready.

Basic Antiderivatives (Day 16)Riemann Sums & Approximation (Day 17)The Definite Integral & the FTC (Day 18)U-Substitution & Accumulation Functions (Day 19)Differential Equations & Slope Fields (Day 20)Exponential Growth & Decay (Day 20)Average Value & Particle Motion (Day 21)

Basic Antiderivatives (Day 16)Forgot it? Go back and study again →

Power rule

∫xn dx=xn+1n+1+C  (n≠−1),∫1x dx=ln⁡∣x∣+C\int x^n\,dx = \frac{x^{n+1}}{n+1} + C \;(n \neq -1), \quad \int \frac{1}{x}\,dx = \ln|x| + C

Exponential & trig

∫ekx dx=1kekx+C,∫sin⁡x dx=−cos⁡x+C,∫cos⁡x dx=sin⁡x+C\int e^{kx}\,dx = \frac{1}{k}e^{kx} + C, \quad \int \sin x\,dx = -\cos x + C, \quad \int \cos x\,dx = \sin x + C
∫sec⁡2x dx=tan⁡x+C,∫sec⁡xtan⁡x dx=sec⁡x+C\int \sec^2 x\,dx = \tan x + C, \quad \int \sec x\tan x\,dx = \sec x + C

Arctangent & arcsine forms

∫dxx2+a2=1atan⁡−1 ⁣(xa)+C,∫dxa2−x2=sin⁡−1 ⁣(xa)+C\int \frac{dx}{x^2 + a^2} = \frac{1}{a}\tan^{-1}\!\left(\frac{x}{a}\right) + C, \quad \int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\!\left(\frac{x}{a}\right) + C

General quadratic denominator: complete the square first.

Initial-value problems

Antidifferentiate, then substitute the given point to solve for CC (twice for a 2nd-order problem).

Riemann Sums & Approximation (Day 17)Forgot it? Go back and study again →

Definite integral as a limit of Riemann sums

∫abf(x) dx=lim⁡n→∞∑i=1nf(xi∗) Δx,Δx=b−an\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}

Left / right / midpoint sums

xi∗=a+(i−1)Δxx_i^* = a + (i-1)\Delta x (left),   a+i Δx\;a + i\,\Delta x (right),   a+(i−12)Δx\;a + (i-\tfrac12)\Delta x (midpoint)

Increasing: left sum under, right sum over. Concave up: trapezoid over, midpoint under.

Trapezoidal rule

∫abf(x) dx≈Δx2[f(x0)+2f(x1)+⋯+2f(xn−1)+f(xn)]\int_a^b f(x)\,dx \approx \frac{\Delta x}{2}\left[f(x_0) + 2f(x_1) + \cdots + 2f(x_{n-1}) + f(x_n)\right]

From a table: widths can be unequal — multiply each piece by its own Δt\Delta t.

The Definite Integral & the FTC (Day 18)Forgot it? Go back and study again →

Properties

∫ab(cf±g)=c∫abf±∫abg,∫acf+∫cbf=∫abf\int_a^b (cf \pm g) = c\int_a^b f \pm \int_a^b g, \quad \int_a^c f + \int_c^b f = \int_a^b f
∫abf=−∫baf,∫aaf=0\int_a^b f = -\int_b^a f, \quad \int_a^a f = 0

FTC Part 1 (evaluation)

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a), where F′=fF' = f

FTC Part 2 (differentiation)

ddx∫au(x)f(t) dt=f(u(x))⋅u′(x)\frac{d}{dx}\int_a^{u(x)} f(t)\,dt = f(u(x)) \cdot u'(x)

If x is the lower limit, the result picks up a minus sign.

U-Substitution & Accumulation Functions (Day 19)Forgot it? Go back and study again →

Substitution rule

∫f(g(x)) g′(x) dx=∫f(u) du,u=g(x),  du=g′(x) dx\int f(g(x))\,g'(x)\,dx = \int f(u)\,du, \quad u = g(x),\; du = g'(x)\,dx

Definite integrals

Change the limits: if u=g(x)u = g(x) then ∫ab→∫g(a)g(b)\int_a^b \to \int_{g(a)}^{g(b)}

Or convert back to x before plugging in the original limits — never mix.

Accumulation function $g(x) = \int_a^x f(t)\,dt$

g′=fg' = f and g′′=f′g'' = f': read gg's increase/extrema off the SIGN of ff; read gg's concavity/inflection off whether ff is INCREASING.

g(a)=0g(a) = 0; g(x)g(x) is signed area under ff from aa to xx.

Differential Equations & Slope Fields (Day 20)Forgot it? Go back and study again →

Separable equation

dydx=f(x)g(y)  ⇒  ∫dyg(y)=∫f(x) dx\frac{dy}{dx} = f(x)g(y) \;\Rightarrow\; \int \frac{dy}{g(y)} = \int f(x)\,dx

Add a single +C on the x side, then use the initial condition to solve for C.

Slope field

At each point (x,y)(x, y) draw a short segment of slope dydx∣(x,y)\frac{dy}{dx}\big|_{(x,y)}

Solution curves flow along the segments; horizontal where dy/dx = 0.

Exponential Growth & Decay (Day 20)Forgot it? Go back and study again →

The model

dydt=ky  ⇒  y(t)=y0ekt,y0=y(0)\frac{dy}{dt} = ky \;\Rightarrow\; y(t) = y_0 e^{kt}, \quad y_0 = y(0)

k > 0 growth, k < 0 decay. Find k from a second data point.

Doubling time / half-life

Doubling time =ln⁡2k,= \frac{\ln 2}{k}, \qquad half-life =ln⁡2∣k∣= \frac{\ln 2}{|k|}

Average Value & Particle Motion (Day 21)Forgot it? Go back and study again →

Average value of a function

favg=1b−a∫abf(x) dxf_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx

MVT for Integrals: some c in (a, b) has f(c) = f_avg.

Displacement vs. total distance

Displacement =∫abv(t) dt= \int_a^b v(t)\,dt, \quad Total distance =∫ab∣v(t)∣ dt= \int_a^b |v(t)|\,dt

For distance, split at every t where v(t) = 0, then add the absolute values.

Position from velocity

s(t)=s(a)+∫atv(τ) dτs(t) = s(a) + \int_a^t v(\tau)\,d\tau

Timed Multiple Choice Quiz

Phase 3 Quiz: Integration

25 timed questions covering everything from Day 16–21. Treat it like a real exam — review your mistakes after.

45 min

Duration

25

Questions

Before you begin

·Pick one answer per question — you can change it any time before submitting.

·The timer starts immediately when you click Start.

·You can submit early at any time.

·Correct answers and explanations only appear after you submit.