Unit 6: Integration and Accumulation of Change

📚Study Guide: Integration and Accumulation of Change

Unit 6: Integration and Accumulation of Change

Integration is the second fundamental pillar of calculus, serving as the inverse operation of differentiation. This unit introduces the concept of accumulation and the definite integral as the limit of Riemann sums. You will learn various techniques of integration including u-substitution, and you will explore the Fundamental Theorem of Calculus (FTC), which connects differential and integral calculus. The FTC allows you to evaluate definite integrals using antiderivatives, dramatically simplifying accumulation problems. You will also study accumulation functions defined by integrals and analyze their properties using calculus. Understanding the conceptual meaning of integration as net accumulation is crucial for applied contexts.

Key Concepts

• Antiderivatives: F is an antiderivative of f if F' = f. The general antiderivative includes +C.
• Riemann Sums: Left, right, midpoint, and trapezoidal approximations estimate definite integrals using finite rectangles or trapezoids.
• Definite Integral: The limit of Riemann sums as n->infinity, representing net signed area under a curve.
• Fundamental Theorem of Calculus Part 1: If F(x) = integral from a to x of f(t) dt, then F'(x) = f(x).
• Fundamental Theorem of Calculus Part 2: Integral from a to b of f(x) dx = F(b) - F(a), where F is any antiderivative of f.
• U-Substitution: A technique for integrating composite functions by substituting u = g(x) and du = g'(x)dx.
• Net Area: Areas above the x-axis are positive; areas below are negative.

Vocabulary

• Antiderivative: A function whose derivative is the given function.
• Integrand: The function being integrated in an integral expression.
• Riemann sum: An approximation of a definite integral using the sum of areas of rectangles or trapezoids.
• Accumulation function: A function defined as the definite integral from a constant to a variable upper limit.
• Fundamental Theorem of Calculus: The theorem linking differentiation and integration as inverse processes.
• Net signed area: The total area between a curve and the x-axis, counting area below the axis as negative.

Formulas

• Integral of x^n dx = x^(n+1)/(n+1) + C, n != -1
• Integral of 1/x dx = ln|x| + C
• Integral of e^x dx = e^x + C
• Integral of sin(x) dx = -cos(x) + C
• Integral of cos(x) dx = sin(x) + C
• FTC 1: d/dx [integral from a to x of f(t) dt] = f(x)
• FTC 2: integral from a to b of f(x) dx = F(b) - F(a)
• U-sub: integral f(g(x))*g'(x) dx = integral f(u) du

Common Mistakes

• Forgetting +C in indefinite integrals; it is required for the general antiderivative.
• Misapplying FTC Part 1 when the upper limit is a function of x, neglecting the chain rule factor.
• Evaluating definite integrals without considering that area below the x-axis subtracts from the total.
• Failing to change the limits of integration when performing u-substitution on a definite integral.

AP Exam Strategies

• For accumulation function derivatives, always check if the upper limit is x or a function of x; if it is a function, multiply by its derivative (chain rule).
• When approximating integrals with Riemann sums, carefully match the function evaluation point to the method (left, right, midpoint).
• On FRQs involving net change, explicitly state that the integral represents accumulation and reference FTC Part 2.
• If an integrand looks like a derivative divided by the function, suspect a natural log antiderivative.

Real-World Applications

• Physics: Displacement is the integral of velocity; total distance is the integral of speed (absolute value of velocity).
• Economics: Consumer and producer surplus are calculated as definite integrals of demand and supply curves.
• Hydrology: Total water flow over time is the integral of the rate of flow function.