AP Calculus AB — Integrals and Applications flashcards

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Units 6–8 of AP Calculus AB: Riemann and trapezoidal sums, accumulation functions, both parts of the Fundamental Theorem, antiderivatives and substitution, slope fields, separable differential equations, average value, motion, area and volume.

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Learn deck · part of AP Calculus AB · written against: College Board AP Calculus AB Course and Exam Description (Units 1–8). Updated 2026-10-10.

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$v(t)$ is a velocity in meters per second. What does $\int_0^{10} v(t) dt$ represent, and how does total distance differ?

It is the net change in position (displacement) in meters over the 10 seconds. Total distance is $\int_0^{10} |v(t)| dt$ ⁠, which counts backward motion as positive.
Integration and Accumulation of Change › Riemann Sums and Accumulation
$R(t)$ is a flow rate in liters per minute and $t$ is in minutes. What are the units of $\int_0^4 R(t) dt$ ⁠?

Liters: the units of a definite integral are the integrand's units times the units of the variable of integration.
Integration and Accumulation of Change › Riemann Sums and Accumulation
For an increasing function, which of the left and right Riemann sums overestimates the integral?

The right Riemann sum overestimates and the left underestimates, because each rectangle's height comes from the higher or lower end of its subinterval. Concavity does not matter for this comparison.
Integration and Accumulation of Change › Riemann Sums and Accumulation
When does a trapezoidal sum overestimate a definite integral?

When the function is concave up: each trapezoid's top edge is a chord lying above the curve. For a concave-down function it underestimates.
Integration and Accumulation of Change › Riemann Sums and Accumulation
How do you verify that $y = e^{2x}$ is a solution of $y'' - 4y = 0$ ⁠?

Substitute it in: $y'' = 4e^{2x}$ ⁠, and $4e^{2x} - 4e^{2x} = 0$ ⁠, so the equation holds for every $x$ ⁠.
Differential Equations › Modeling and Slope Fields
What is the difference between a general and a particular solution of a differential equation?

A general solution is a family containing an arbitrary constant; a particular solution is the one member picked out by an initial condition.
Differential Equations › Modeling and Slope Fields
A slope field's segments are identical along every vertical line. What does that tell you about $\frac{dy}{dx}$ ⁠?

The slope depends only on $x$ ⁠. If segments are identical along every horizontal line instead, the slope depends only on $y$ ⁠.
Differential Equations › Modeling and Slope Fields
What is the average value of $f$ on $[a, b]$ ⁠?

$\frac{1}{b - a} \int_a^b f(x) dx$ ⁠: the height of the rectangle on $[a, b]$ with the same area as the region under $f$ ⁠.
Applications of Integration › Average Value and Motion
For $f(x) = x^2$ on $[0, 6]$ ⁠, what are the average value and the average rate of change?

Average value $\frac{1}{6}\int_0^6 x^2 dx = 12$ ⁠; average rate of change $\frac{36 - 0}{6} = 6$ ⁠. One averages $f$ ⁠, the other averages $f'$ ⁠.
Applications of Integration › Average Value and Motion
How do you find a particle's position at time $b$ from its position at time $a$ and its velocity?

$x(b) = x(a) + \int_a^b v(t) dt$ ⁠: starting position plus displacement.
Applications of Integration › Average Value and Motion

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Topic outline follows the College Board AP Calculus AB and BC Course and Exam Description (effective Fall 2020), Units 6–8 (the AP name is used nominatively; no course or exam text reproduced).
OpenStax Calculus Volumes 1 and 2, consulted as references; no text copied.

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