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Day 22 · Review + Quiz

Phase 3 Review

Every formula from Day 18–21 in one place. Skim it, then take the quiz below when you're ready.

Riemann Sums & the Definite Integral (Day 18)Antiderivatives & the Fundamental Theorem (Day 19)U-Substitution (Day 19)Differential Equations & Slope Fields (Day 20)Exponential Growth & Decay (Day 20)Average Value & Particle Motion (Day 21)

Riemann Sums & the Definite Integral (Day 18)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 function: left sum underestimates, right sum overestimates. Decreasing: reversed.

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]

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

Antiderivatives & the Fundamental Theorem (Day 19)Forgot it? Go back and study again →

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.

Power rule for antiderivatives

∫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

Common antiderivatives

∫ex dx=ex+C,∫sin⁡x dx=−cos⁡x+C,∫cos⁡x dx=sin⁡x+C\int e^x\,dx = e^x + C, \quad \int \sin x\,dx = -\cos x + C, \quad \int \cos x\,dx = \sin x + C
∫sec⁡2x dx=tan⁡x+C,∫11+x2 dx=tan⁡−1x+C\int \sec^2 x\,dx = \tan x + C, \quad \int \frac{1}{1+x^2}\,dx = \tan^{-1}x + C

U-Substitution (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.

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 18–21. 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.