Rucete ✏ AP Calculus In a Nutshell
6. Definite Integrals
This chapter covers the evaluation and interpretation of definite integrals using analytical, graphical, and numerical methods. It emphasizes the Fundamental Theorem of Calculus and introduces Riemann sums, trapezoidal approximations, and average value concepts. BC topics include parametric integrals.
Fundamental Theorem of Calculus (FTC)
• If f is continuous on [a, b] and F′(x) = f(x), then:
∫ab f(x) dx = F(b) – F(a)
• This gives the net change in F from a to b
• Any continuous, bounded function on [a, b] is integrable
Properties of Definite Integrals
• Linearity: ∫ab [cf(x) + g(x)] dx = c∫ab f(x) dx + ∫ab g(x) dx
• Reversal: ∫ba f(x) dx = –∫ab f(x) dx
• Additivity: ∫ac f(x) dx + ∫cb f(x) dx = ∫ab f(x) dx
Mean Value Theorem for Integrals
• If f is continuous on [a, b], then there exists c ∈ (a, b) such that:
∫ab f(x) dx = f(c)(b – a)
Definite Integrals from Riemann Sums
• A definite integral is the limit of a Riemann sum
• Divide [a, b] into n equal subintervals of width Δx = (b – a)/n
• Choose sample points xk in each interval
• Riemann sum: ∑ f(xk)Δx → ∫ab f(x) dx as n → ∞
Left, Right, and Midpoint Sums
• Left sum (Ln): use left endpoint of each subinterval • Right sum (Rn): use right endpoint of each subinterval • Midpoint sum (Mn): use midpoint of each subinterval
• If f is increasing: Ln < true value < Rn
• If f is decreasing: Ln > true value > Rn
Trapezoidal Approximation
• Uses trapezoids instead of rectangles • Formula: Tn = (Δx / 2) [f(x0) + 2f(x1) + ... + 2f(xn–1) + f(xn)]
• Usually more accurate than Ln and Rn
Average Value of a Function
• If f is integrable on [a, b], the average value is:
favg = (1 / (b – a)) ∫ab f(x) dx
• This gives the height of a rectangle with the same area under f on [a, b]
Applications of FTC
• If G(x) = ∫ax f(t) dt, then G′(x) = f(x)
• This defines accumulation functions, showing how area builds up as x increases
• If the lower limit is a variable: d/dx [∫xb f(t) dt] = –f(x)
• If limits are both functions: d/dx [∫g(x)h(x) f(t) dt] = f(h(x))h′(x) – f(g(x))g′(x)
Area Under a Curve
• Area between f(x) and x-axis on [a, b]: ∫ab |f(x)| dx
• Area between f(x) and g(x): ∫ab |f(x) – g(x)| dx → Use f(x) – g(x) if f ≥ g throughout
ln(x) as an Integral
• ln(x) = ∫1x (1 / t) dt • This definition can be used for derivatives and inverse rules
Using a Graph to Interpret an Integral
• Use geometry (triangle, trapezoid, rectangle) when exact graphs are provided
• Watch for regions below the x-axis (those contribute negative area)
• Area between curves can also be estimated visually
Distance Traveled and Total Accumulation
• If v(t) is velocity, total displacement: ∫ab v(t) dt
• Total distance: ∫ab |v(t)| dt
• Always use absolute value for total distance
Integrals of Parametric Curves (BC)
• For x = f(t), y = g(t): Arc length = ∫ab √[(dx/dt)² + (dy/dt)²] dt
In a Nutshell
Definite integrals quantify net change, total area, or accumulated values. The Fundamental Theorem connects derivatives and integrals, while Riemann sums and trapezoids offer numerical approximations. Applications include motion, average values, parametric curves (BC), and interpreting area under or between curves.