Accumulation Functions, Definite Intervals, Applied Contexts - AP Calculus BC
Card 1 of 30
Determine $\int_{-1}^1 x^3,dx$.
Determine $\int_{-1}^1 x^3,dx$.
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$\int_{-1}^1 x^3,dx = 0$. Odd function over symmetric interval cancels.
$\int_{-1}^1 x^3,dx = 0$. Odd function over symmetric interval cancels.
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Express the net change of a quantity using integrals.
Express the net change of a quantity using integrals.
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Net change is $\int_a^b F'(x),dx = F(b) - F(a)$. FTC relates rate of change to total change.
Net change is $\int_a^b F'(x),dx = F(b) - F(a)$. FTC relates rate of change to total change.
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State the integration property for $\int_a^b [f(x) + g(x)],dx$.
State the integration property for $\int_a^b [f(x) + g(x)],dx$.
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$\int_a^b f(x),dx + \int_a^b g(x),dx$. Linearity property of integrals.
$\int_a^b f(x),dx + \int_a^b g(x),dx$. Linearity property of integrals.
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What is the formula for accumulation function $A(x)$?
What is the formula for accumulation function $A(x)$?
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$A(x) = \int_a^x f(t),dt$ where $a$ is a constant and $f(t)$ is a given function. Accumulates $f(t)$ from fixed point $a$ to variable $x$.
$A(x) = \int_a^x f(t),dt$ where $a$ is a constant and $f(t)$ is a given function. Accumulates $f(t)$ from fixed point $a$ to variable $x$.
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Evaluate the integral of a constant, $\int_a^b c,dx$.
Evaluate the integral of a constant, $\int_a^b c,dx$.
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$c(b-a)$, where $c$ is a constant. Constant times interval length.
$c(b-a)$, where $c$ is a constant. Constant times interval length.
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Evaluate $\int_0^4 (x^3 - 2x),dx$.
Evaluate $\int_0^4 (x^3 - 2x),dx$.
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$\int_0^4 (x^3 - 2x),dx = 48$. Using $[\frac{x^4}{4} - x^2]_0^4 = 64 - 16$.
$\int_0^4 (x^3 - 2x),dx = 48$. Using $[\frac{x^4}{4} - x^2]_0^4 = 64 - 16$.
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Calculate $\int_0^2 (2x + 1),dx$.
Calculate $\int_0^2 (2x + 1),dx$.
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$\int_0^2 (2x + 1),dx = 6$. Antiderivative: $[x^2 + x]_0^2 = 4 + 2 = 6$.
$\int_0^2 (2x + 1),dx = 6$. Antiderivative: $[x^2 + x]_0^2 = 4 + 2 = 6$.
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Determine the value of $\int_1^3 x^2,dx$.
Determine the value of $\int_1^3 x^2,dx$.
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$\int_1^3 x^2,dx = 26/3$. Using $[\frac{x^3}{3}]_1^3 = 9 - \frac{1}{3}$.
$\int_1^3 x^2,dx = 26/3$. Using $[\frac{x^3}{3}]_1^3 = 9 - \frac{1}{3}$.
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What does the definite integral $\int_a^b r(t),dt$ represent in economics?
What does the definite integral $\int_a^b r(t),dt$ represent in economics?
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Total accumulated revenue over time from $a$ to $b$. Integrating revenue rate gives total revenue.
Total accumulated revenue over time from $a$ to $b$. Integrating revenue rate gives total revenue.
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What does $\int_0^T f(t),dt$ represent in population models?
What does $\int_0^T f(t),dt$ represent in population models?
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Total population change over time $0$ to $T$. Integrating population growth rate.
Total population change over time $0$ to $T$. Integrating population growth rate.
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What is the integral of a function over an interval with zero length?
What is the integral of a function over an interval with zero length?
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Zero, since $\int_a^a f(x),dx = 0$. Degenerate interval has no area.
Zero, since $\int_a^a f(x),dx = 0$. Degenerate interval has no area.
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Evaluate $\int_1^2 \frac{1}{x},dx$.
Evaluate $\int_1^2 \frac{1}{x},dx$.
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$\int_1^2 \frac{1}{x},dx = \ln 2$. Natural logarithm is antiderivative of $\frac{1}{x}$.
$\int_1^2 \frac{1}{x},dx = \ln 2$. Natural logarithm is antiderivative of $\frac{1}{x}$.
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Determine the integral $\int_0^3 4x,dx$.
Determine the integral $\int_0^3 4x,dx$.
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$\int_0^3 4x,dx = 18$. Using $[2x^2]_0^3 = 2 \cdot 9$.
$\int_0^3 4x,dx = 18$. Using $[2x^2]_0^3 = 2 \cdot 9$.
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Find the value of $\int_0^2 (2 + x),dx$.
Find the value of $\int_0^2 (2 + x),dx$.
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$\int_0^2 (2 + x),dx = 6$. Using $[2x + \frac{x^2}{2}]_0^2$.
$\int_0^2 (2 + x),dx = 6$. Using $[2x + \frac{x^2}{2}]_0^2$.
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What is the accumulation function for $f(t) = \cos t$ from $a = 0$?
What is the accumulation function for $f(t) = \cos t$ from $a = 0$?
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$A(x) = \int_0^x \cos t,dt = \sin x$. Using $[\sin t]_0^x = \sin x - 0$.
$A(x) = \int_0^x \cos t,dt = \sin x$. Using $[\sin t]_0^x = \sin x - 0$.
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What is the geometric interpretation of $\int_a^b f(x),dx$ when $f(x) \geq 0$?
What is the geometric interpretation of $\int_a^b f(x),dx$ when $f(x) \geq 0$?
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It is the area under the curve $f(x)$ from $x = a$ to $x = b$. Region bounded by curve and x-axis.
It is the area under the curve $f(x)$ from $x = a$ to $x = b$. Region bounded by curve and x-axis.
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What does the Fundamental Theorem of Calculus Part 2 state?
What does the Fundamental Theorem of Calculus Part 2 state?
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If $f$ is continuous on $[a, b]$, $F(x) = \int_a^x f(t),dt$ is continuous and differentiable. Shows accumulation functions are differentiable.
If $f$ is continuous on $[a, b]$, $F(x) = \int_a^x f(t),dt$ is continuous and differentiable. Shows accumulation functions are differentiable.
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Identify the expression for the average value of $f(x)$ on $[a, b]$.
Identify the expression for the average value of $f(x)$ on $[a, b]$.
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The average value is $\frac{1}{b-a} \int_a^b f(x),dx$. Divides total area by interval width.
The average value is $\frac{1}{b-a} \int_a^b f(x),dx$. Divides total area by interval width.
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How do you interpret $\int_a^b v(t),dt$ in a physics context?
How do you interpret $\int_a^b v(t),dt$ in a physics context?
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It represents the displacement of an object from time $a$ to $b$ given velocity $v(t)$.. Integrating velocity gives position change.
It represents the displacement of an object from time $a$ to $b$ given velocity $v(t)$.. Integrating velocity gives position change.
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How does $\int_a^b f(x),dx$ change if $a$ and $b$ are swapped?
How does $\int_a^b f(x),dx$ change if $a$ and $b$ are swapped?
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It becomes $-\int_b^a f(x),dx$. Reversing limits changes sign.
It becomes $-\int_b^a f(x),dx$. Reversing limits changes sign.
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What is the Fundamental Theorem of Calculus Part 1?
What is the Fundamental Theorem of Calculus Part 1?
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If $F$ is an antiderivative of $f$ on $[a, b]$, then $\int_a^b f(x),dx = F(b) - F(a)$. Connects antiderivatives to definite integrals.
If $F$ is an antiderivative of $f$ on $[a, b]$, then $\int_a^b f(x),dx = F(b) - F(a)$. Connects antiderivatives to definite integrals.
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What is the integral of $\int_a^b kf(x),dx$ where $k$ is constant?
What is the integral of $\int_a^b kf(x),dx$ where $k$ is constant?
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$k \int_a^b f(x),dx$. Constant factor property of integrals.
$k \int_a^b f(x),dx$. Constant factor property of integrals.
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How do you calculate the total distance traveled from $t = a$ to $t = b$?
How do you calculate the total distance traveled from $t = a$ to $t = b$?
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$\int_a^b |v(t)|,dt$ for velocity function $v(t)$.. Absolute value ensures positive distances.
$\int_a^b |v(t)|,dt$ for velocity function $v(t)$.. Absolute value ensures positive distances.
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Find the accumulation function $A(x)$ given $f(t) = 3t^2$ and $a = 1$.
Find the accumulation function $A(x)$ given $f(t) = 3t^2$ and $a = 1$.
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$A(x) = \int_1^x 3t^2,dt = x^3 - 1$. Using FTC: $\int_1^x 3t^2,dt = [t^3]_1^x$.
$A(x) = \int_1^x 3t^2,dt = x^3 - 1$. Using FTC: $\int_1^x 3t^2,dt = [t^3]_1^x$.
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Find $\int_0^2 (x^2 + 3),dx$.
Find $\int_0^2 (x^2 + 3),dx$.
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$\int_0^2 (x^2 + 3),dx = \frac{26}{3}$. Using $[\frac{x^3}{3} + 3x]_0^2$.
$\int_0^2 (x^2 + 3),dx = \frac{26}{3}$. Using $[\frac{x^3}{3} + 3x]_0^2$.
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What is $\int_a^a f(x),dx$ equal to?
What is $\int_a^a f(x),dx$ equal to?
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Zero, since the interval has zero length. No area when endpoints are identical.
Zero, since the interval has zero length. No area when endpoints are identical.
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What is the result of $\int_0^1 5,dx$?
What is the result of $\int_0^1 5,dx$?
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5, since the integral of a constant is the constant times the interval length. Constant $5$ over interval length $1$.
5, since the integral of a constant is the constant times the interval length. Constant $5$ over interval length $1$.
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State the relationship between definite integral and area under a curve.
State the relationship between definite integral and area under a curve.
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The definite integral $\int_a^b f(x),dx$ represents the net area between $f(x)$ and the x-axis. Positive areas above axis, negative below.
The definite integral $\int_a^b f(x),dx$ represents the net area between $f(x)$ and the x-axis. Positive areas above axis, negative below.
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Find $A(x)$ given $f(t) = e^t$ and $a = 0$.
Find $A(x)$ given $f(t) = e^t$ and $a = 0$.
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$A(x) = \int_0^x e^t,dt = e^x - 1$. Using $[e^t]_0^x = e^x - e^0$.
$A(x) = \int_0^x e^t,dt = e^x - 1$. Using $[e^t]_0^x = e^x - e^0$.
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Express $\int_a^b f(x),dx$ using the limit of sums.
Express $\int_a^b f(x),dx$ using the limit of sums.
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$\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$. Riemann sum definition of definite integral.
$\lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$. Riemann sum definition of definite integral.
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