Stewart Calculus · 8th Edition
Chapter 5: Integrals
Every key formula from Stewart Calculus Chapter 5, in one searchable page. Click a card to study it — worked examples included.
5.1 · AREAS & DISTANCES
Riemann sums
Split [a, b] into n strips of width Δx = b − an; add rectangle areas f(xi*)Δx.
A ≈ nΣi = 1 f(xi*)Δx
Left / right / midpoint = where you sample xi* in each strip.
WORKED EXAMPLE
∫02 x² dx with R₄: Δx = 0.5, heights 0.25, 1, 2.25, 4.
R₄ = 0.5(0.25 + 1 + 2.25 + 4) = 0.5(7.5) = 3.75.
R₄ = 0.5(0.25 + 1 + 2.25 + 4) = 0.5(7.5) = 3.75.
KEY NOTES
- For increasing f: right endpoints overestimate, left underestimate.
5.2 · THE DEFINITE INTEGRAL
Definition (limit of Riemann sums)
∫ab f(x) dx = limn→∞ nΣi = 1 f(xi*)Δx
Net signed area: above the axis +, below the axis −.
Properties
∫ab cf = c∫ab f · ∫ab(f ± g) = ∫ab f ± ∫ab g
∫ab f = −∫ba f · ∫ac f + ∫cb f = ∫ab f
f ≤ g on [a,b] ⇒ ∫ab f ≤ ∫ab g · ∫aa f = 0
∫ab f = −∫ba f · ∫ac f + ∫cb f = ∫ab f
f ≤ g on [a,b] ⇒ ∫ab f ≤ ∫ab g · ∫aa f = 0
5.3 · FUNDAMENTAL THEOREM
Part 1 — differentiation undoes integration
g(x) = ∫ax f(t) dt ⇒ g′(x) = f(x)
Part 2 — evaluate with antiderivatives
∫ab f(x) dx = F(b) − F(a), where F′ = f
WORKED EXAMPLE
∫03 (2x + 1) dx = [x² + x]03 = (9 + 3) − 0 = 12.
WORKED EXAMPLE
ddx ∫0x² sin t dt = sin(x²) · 2x = 2x sin(x²) (chain rule!)
KEY NOTES
- Upper limit u(x) ⇒ multiply by u′(x) (chain rule).
5.4 · INDEFINITE & NET CHANGE
Indefinite integral
∫ f(x) dx = F(x) + C, where F′ = f
xn → xn+1n+1 (n ≠ −1) · 1x → ln|x| · ex → ex
sin x → −cos x · cos x → sin x
sin x → −cos x · cos x → sin x
Net Change Theorem
∫ab F′(x) dx = F(b) − F(a)
displacement = ∫ab v dt · distance = ∫ab |v| dt
5.5 · SUBSTITUTION RULE
The method (u-substitution)
- Let u = g(x) (the “inside” function)
- Compute du = g′(x) dx — it must appear
- Rewrite as ∫ f(u) du, integrate, substitute back
WORKED EXAMPLE
∫ xdx: u = x² + 1, du = 2x dx.
12∫ u½ du = 12 · 23u3/2 = (x²+1)3/23 + C
12∫ u½ du = 12 · 23u3/2 = (x²+1)3/23 + C
KEY NOTES
- Definite ∫ab: change limits — u = g(a) to u = g(b).
SIGMA & INTEGRAL TOOLKIT
Summation formulas
nΣi=1 c = cn · nΣi=1 i = n(n+1)2
nΣi=1 i² = n(n+1)(2n+1)6 · nΣi=1 i³ = [n(n+1)2]²
nΣi=1 i² = n(n+1)(2n+1)6 · nΣi=1 i³ = [n(n+1)2]²
PITFALLS
- +C belongs ONLY on indefinite integrals.
- The variable of integration is a dummy: ∫ab f(x)dx = ∫ab f(t)dt.
WATCH OUT!
Mistakes that cost points
- ∫ dx/x = ln|x| + C — power rule fails at n=−1
- ∫f·g ≠ (∫f)(∫g) — integrals don’t split products
- Never forget +C on indefinite integrals
- ∫ab f = F(b) − F(a) — always top minus bottom
- Split at zeros: ∫|v|dt needs v = 0 points
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