AP Calculus AB Unit 6 study games — Integration and Accumulation.
This unit covers Riemann sums, Fundamental Theorem and antiderivatives — essential concepts for AP Calculus AB. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers Riemann sums, Fundamental Theorem and antiderivatives — essential concepts for AP Calculus AB. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Riemann Sums
Riemann sums approximate the definite integral by dividing an interval into subintervals and summing the areas of rectangles. Students must be able to set up left, right, and midpoint Riemann sums from a table of values or a graph. On the exam, you may also need to determine whether a sum over- or under-estimates the integral based on the function's behavior.
Key Points
- Left Riemann sum uses the left endpoint of each subinterval; overestimates when f is decreasing, underestimates when increasing
- Right Riemann sum uses the right endpoint; overestimates when f is increasing, underestimates when decreasing
- Midpoint Riemann sum generally gives a better approximation and is tested with tables of values
- Trapezoidal sum averages left and right: T = (Δx/2)[f(x₀) + 2f(x₁) + ... + 2f(x_{n-1}) + f(xₙ)]
The table gives values of a continuous function f: x = 0, 2, 4, 6 with f(x) = 3, 7, 4, 9. Using a right Riemann sum with 3 equal subintervals, approximate ∫₀⁶ f(x) dx.
Each subinterval has width Δx = 2. The right Riemann sum uses the right endpoint of each subinterval: f(2), f(4), f(6). The approximation is 2·(7 + 4 + 9) = 2·20 = 40.
2 Fundamental Theorem of Calculus
The FTC has two parts that are both tested heavily. Part 1 states that if g(x) = ∫ₐˣ f(t) dt, then g′(x) = f(x); students must apply the chain rule when the upper limit is a function of x. Part 2 states that ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f, and is the basis for evaluating all definite integrals.
Key Points
- FTC Part 1: d/dx[∫ₐˣ f(t) dt] = f(x); if upper limit is g(x), multiply by g′(x)
- FTC Part 2: ∫ₐᵇ f(x) dx = F(b) − F(a); order of limits matters — swapping limits negates the integral
- ∫ₐᵃ f(x) dx = 0 always; ∫ₐᵇ f(x) dx = −∫ᵦᵃ f(x) dx
- Accumulation function g(x) = ∫ₐˣ f(t) dt: g is increasing where f > 0, g has a local max/min where f changes sign
Let g(x) = ∫₁^(x²) sin(t) dt. Find g′(x).
By FTC Part 1 with the chain rule, g′(x) = sin(x²) · d/dx(x²). Differentiating the upper limit gives 2x. Therefore g′(x) = 2x sin(x²).
3 Antiderivatives
An antiderivative F of f satisfies F′(x) = f(x); the general antiderivative always includes +C for indefinite integrals. Students must know the standard antiderivative rules and be able to use u-substitution to handle composite functions. Initial conditions are used on the exam to solve for C and write a specific antiderivative.
Key Points
- Core rules: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫eˣ dx = eˣ + C, ∫(1/x) dx = ln|x| + C, ∫cos x dx = sin x + C, ∫sin x dx = −cos x + C
- u-substitution: identify an inner function u = g(x), compute du = g′(x) dx, rewrite the integral entirely in terms of u before integrating
- For definite integrals with u-sub, either convert the limits to u-values or back-substitute before evaluating
- Given f′(x) and an initial condition f(a) = b, integrate f′ then solve for C using the given point
Find the particular solution to dy/dx = 3x² − 6x given that y(1) = 4.
Integrate: y = x³ − 3x² + C. Apply the initial condition: 4 = (1)³ − 3(1)² + C = 1 − 3 + C, so C = 6. The particular solution is y = x³ − 3x² + 6.
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What is Integration and Accumulation?
Integration and Accumulation is Unit 6 of AP Calculus AB, covering Riemann sums, Fundamental Theorem and antiderivatives.
How to study for AP Calculus AB Unit 6?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 30+ review questions across 5 different game modes.