Chapter 3: Integration

3.1 Integration as the Inverse of Differentiation

• Integration is the reverse process of differentiation. (p.83)
• If d/dx[f(x)] = f′(x), then ∫ f′(x) dx = f(x). (p.83)

3.2 Indefinite Integral

• ∫ a dx = ax + c (a, c constants). (p.85)
• ∫ axⁿ dx = axⁿ⁺¹/(n + 1) + c, n ≠ −1. (p.85)
• Steps: add 1 to the index of x, divide by the new index, add the constant c. (p.85)
• The constant c is the constant of integration, since differentiating gives the same dy/dx for different constants. (p.86)
• ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx; ∫ axⁿ dx = a ∫ xⁿ dx. (p.86)
• ∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / [a(n + 1)] + c, n ≠ −1. (p.88)
• Substitution method: let u = ax + b, so dx = du/a. (p.89)
• Given the gradient function dy/dx = f(x), the equation of the curve is y = ∫ f(x) dx. (p.90)
• Find c by substituting a known point (x, y) into the integrated equation. (p.90)

3.3 Definite Integral

• ∫_a^b f(x) dx = [g(x)]_a^b = g(b) − g(a). (p.92)
• ∫_a^a f(x) dx = 0. (p.94)
• ∫_a^b f(x) dx = −∫_b^a f(x) dx. (p.94)
• ∫_a^b kf(x) dx = k∫_a^b f(x) dx. (p.94)
• ∫_a^b f(x) dx + ∫_b^c f(x) dx = ∫_a^c f(x) dx. (p.94)
• ∫_a^b [f(x) ± g(x)] dx = ∫_a^b f(x) dx ± ∫_a^b g(x) dx. (p.94)

Area of a Region

• As the number of thin rectangles increases, their total area approaches the actual area under the curve. (p.96)
• Area under the curve = lim_δx→0 Σ y_iδx = ∫_a^b y dx. (p.96)
• Region below the x-axis gives a negative integral; above gives a positive integral; the actual area is always positive. (p.97)
• If a curve crosses the x-axis, find the regions above and below separately and add their positive areas. (p.98)
• Area between the curve and the y-axis: A = ∫_a^b x dy. (p.99)
• Region to the right of the y-axis gives a positive integral; to the left gives a negative integral. (p.99)
• Area between a curve y = g(x) and a line y = f(x) = ∫_a^b [upper − lower] dx. (p.102)
• To find where a line and a curve intersect, substitute one equation into the other. (p.103)
• Area between two curves = ∫ (upper curve − lower curve) dx. (p.104)

Volume of Revolution

• A solid is generated when a region is revolved through 360° about an axis. (p.105)
• About the x-axis: V = ∫_a^b πy² dx. (p.107)
• About the y-axis: V = ∫_a^b πx² dy. (p.107)
• The generated volume is always positive. (p.106)
• For a region between a line and a curve, subtract the two generated volumes (V₁ − V₂). (p.109)

3.4 Applications of Integration

• Integration is used to find areas, volumes and quantities from rates of change (e.g. growth of a bacteria colony, dA/dt). (p.111)
• For a parabolic bowl y = ax², substitute a known point to find a, then use ∫πx² dy for the internal volume. (p.111)
• Total quantity from a rate = ∫ (rate) dt over the given time interval. (p.112)