Fundamental Theorem of Calculus: Accumulation Functions - AP Calculus AB
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What is the integral of $e^x$ from $a$ to $b$?
What is the integral of $e^x$ from $a$ to $b$?
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$\int_{a}^{b} e^x , dx = e^b - e^a$. Standard exponential function integration formula.
$\int_{a}^{b} e^x , dx = e^b - e^a$. Standard exponential function integration formula.
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Find $\int_{-2}^{2} x^3 , dx$.
Find $\int_{-2}^{2} x^3 , dx$.
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- $x^3$ is odd, so integral over symmetric interval is zero.
- $x^3$ is odd, so integral over symmetric interval is zero.
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Determine $\int_{0}^{1} (2x^3 + 3x^2) , dx$.
Determine $\int_{0}^{1} (2x^3 + 3x^2) , dx$.
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$\frac{5}{2}$. Antiderivative is $\frac{x^4}{2} + x^3$; evaluate at limits: $\frac{5}{2}$.
$\frac{5}{2}$. Antiderivative is $\frac{x^4}{2} + x^3$; evaluate at limits: $\frac{5}{2}$.
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Identify the error: $\int_{a}^{b} f(x) , dx = F(b) + F(a)$ where $F$ is an antiderivative of $f$.
Identify the error: $\int_{a}^{b} f(x) , dx = F(b) + F(a)$ where $F$ is an antiderivative of $f$.
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Correct: $\int_{a}^{b} f(x) , dx = F(b) - F(a)$. Should subtract, not add: $F(b) - F(a)$.
Correct: $\int_{a}^{b} f(x) , dx = F(b) - F(a)$. Should subtract, not add: $F(b) - F(a)$.
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Evaluate $\int_{1}^{4} \frac{1}{x} , dx$.
Evaluate $\int_{1}^{4} \frac{1}{x} , dx$.
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$\ln 4$. Antiderivative of $\frac{1}{x}$ is $\ln|x|$; evaluate: $\ln(4) - \ln(1) = \ln(4)$.
$\ln 4$. Antiderivative of $\frac{1}{x}$ is $\ln|x|$; evaluate: $\ln(4) - \ln(1) = \ln(4)$.
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What is the result of $\int_{a}^{b} f(x) , dx = 0$?
What is the result of $\int_{a}^{b} f(x) , dx = 0$?
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The areas above and below the x-axis are equal. Net area is zero when positive and negative areas cancel.
The areas above and below the x-axis are equal. Net area is zero when positive and negative areas cancel.
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Evaluate $\int_{0}^{\pi} \sin x , dx$.
Evaluate $\int_{0}^{\pi} \sin x , dx$.
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- Antiderivative of $\sin x$ is $-\cos x$; evaluate: $-\cos(\pi) - (-\cos(0)) = 2$.
- Antiderivative of $\sin x$ is $-\cos x$; evaluate: $-\cos(\pi) - (-\cos(0)) = 2$.
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What is $\frac{d}{dx} \int_{x}^{1} t^2 , dt$?
What is $\frac{d}{dx} \int_{x}^{1} t^2 , dt$?
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$-x^2$. Swapping limits introduces negative sign by FTC.
$-x^2$. Swapping limits introduces negative sign by FTC.
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Evaluate $\int_{0}^{2} x , dx$ using the Fundamental Theorem of Calculus.
Evaluate $\int_{0}^{2} x , dx$ using the Fundamental Theorem of Calculus.
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- Antiderivative is $\frac{x^2}{2}$; evaluate: $\frac{4}{2} - 0 = 2$.
- Antiderivative is $\frac{x^2}{2}$; evaluate: $\frac{4}{2} - 0 = 2$.
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Evaluate $\int_{0}^{4} (3x^2) , dx$ using the Fundamental Theorem of Calculus.
Evaluate $\int_{0}^{4} (3x^2) , dx$ using the Fundamental Theorem of Calculus.
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- Antiderivative is $x^3$; evaluate: $64 - 0 = 64$.
- Antiderivative is $x^3$; evaluate: $64 - 0 = 64$.
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State the effect of swapping integration limits on the integral value.
State the effect of swapping integration limits on the integral value.
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It negates the integral. Swapping limits introduces a negative sign.
It negates the integral. Swapping limits introduces a negative sign.
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What is the value of $\int_{-a}^{a} f(x) , dx$ if $f$ is an odd function?
What is the value of $\int_{-a}^{a} f(x) , dx$ if $f$ is an odd function?
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- Odd functions have symmetric areas that cancel over symmetric intervals.
- Odd functions have symmetric areas that cancel over symmetric intervals.
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What is the geometric interpretation of $\int_{a}^{b} f(x) , dx$?
What is the geometric interpretation of $\int_{a}^{b} f(x) , dx$?
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Net area between $f(x)$ and the x-axis from $x=a$ to $x=b$. Signed area between curve and x-axis over the interval.
Net area between $f(x)$ and the x-axis from $x=a$ to $x=b$. Signed area between curve and x-axis over the interval.
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Evaluate $\int_{0}^{3} (4x + 1) , dx$ using the Fundamental Theorem of Calculus.
Evaluate $\int_{0}^{3} (4x + 1) , dx$ using the Fundamental Theorem of Calculus.
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$\frac{45}{2}$. Find antiderivative $2x^2 + x$, evaluate at limits: $(18+3) - (0) = \frac{45}{2}$.
$\frac{45}{2}$. Find antiderivative $2x^2 + x$, evaluate at limits: $(18+3) - (0) = \frac{45}{2}$.
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What is $\int_{a}^{b} [f(x) + g(x)] , dx$?
What is $\int_{a}^{b} [f(x) + g(x)] , dx$?
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$\int_{a}^{b} f(x) , dx + \int_{a}^{b} g(x) , dx$. Linearity property of integration.
$\int_{a}^{b} f(x) , dx + \int_{a}^{b} g(x) , dx$. Linearity property of integration.
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What is the Fundamental Theorem of Calculus, Part 2?
What is the Fundamental Theorem of Calculus, Part 2?
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If $f$ is continuous on $[a, b]$, then $g(x) = \int_{a}^{x} f(t) , dt$ is differentiable and $g'(x) = f(x)$. Shows that differentiation undoes integration for accumulation functions.
If $f$ is continuous on $[a, b]$, then $g(x) = \int_{a}^{x} f(t) , dt$ is differentiable and $g'(x) = f(x)$. Shows that differentiation undoes integration for accumulation functions.
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Determine $\int_{1}^{2} (2x - 1) , dx$ using an antiderivative.
Determine $\int_{1}^{2} (2x - 1) , dx$ using an antiderivative.
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- Antiderivative is $x^2 - x$; evaluate: $(4-2) - (1-1) = 1$.
- Antiderivative is $x^2 - x$; evaluate: $(4-2) - (1-1) = 1$.
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What is the integral of $x^n$ from $a$ to $b$ for $n \neq -1$?
What is the integral of $x^n$ from $a$ to $b$ for $n \neq -1$?
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$\int_{a}^{b} x^n , dx = \frac{1}{n+1}(b^{n+1} - a^{n+1})$. Apply power rule for integration to definite integrals.
$\int_{a}^{b} x^n , dx = \frac{1}{n+1}(b^{n+1} - a^{n+1})$. Apply power rule for integration to definite integrals.
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Find $\frac{d}{dx} \int_{0}^{x} e^t , dt$.
Find $\frac{d}{dx} \int_{0}^{x} e^t , dt$.
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$e^x$. By FTC Part 2, derivative equals the integrand.
$e^x$. By FTC Part 2, derivative equals the integrand.
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What is the meaning of $\int_{a}^{b} f(x) , dx = \int_{a}^{c} f(x) , dx + \int_{c}^{b} f(x) , dx$?
What is the meaning of $\int_{a}^{b} f(x) , dx = \int_{a}^{c} f(x) , dx + \int_{c}^{b} f(x) , dx$?
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Additivity over intervals. Integrals can be split at any intermediate point.
Additivity over intervals. Integrals can be split at any intermediate point.
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What property does $\int_{a}^{b} f(x) , dx = -\int_{b}^{a} f(x) , dx$ illustrate?
What property does $\int_{a}^{b} f(x) , dx = -\int_{b}^{a} f(x) , dx$ illustrate?
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Reversal of limits changes the sign. Fundamental property showing orientation matters.
Reversal of limits changes the sign. Fundamental property showing orientation matters.
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State the relationship between definite integrals and net area.
State the relationship between definite integrals and net area.
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Definite integrals represent the net area under a curve. Positive area above x-axis minus negative area below.
Definite integrals represent the net area under a curve. Positive area above x-axis minus negative area below.
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Find $\frac{d}{dx} \int_{a}^{g(x)} f(t) , dt$ using the chain rule.
Find $\frac{d}{dx} \int_{a}^{g(x)} f(t) , dt$ using the chain rule.
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$f(g(x))g'(x)$. Chain rule applied to variable upper limit of integration.
$f(g(x))g'(x)$. Chain rule applied to variable upper limit of integration.
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What does the Fundamental Theorem of Calculus connect?
What does the Fundamental Theorem of Calculus connect?
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Differentiation and integration. The FTC bridges these two fundamental operations.
Differentiation and integration. The FTC bridges these two fundamental operations.
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What condition must a function $f$ meet to use the Fundamental Theorem of Calculus?
What condition must a function $f$ meet to use the Fundamental Theorem of Calculus?
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$f$ must be continuous on $[a, b]$. Continuity ensures the theorem applies.
$f$ must be continuous on $[a, b]$. Continuity ensures the theorem applies.
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State the formula for the derivative of an accumulation function $G(x) = \int_{a}^{x} f(t) , dt$.
State the formula for the derivative of an accumulation function $G(x) = \int_{a}^{x} f(t) , dt$.
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$G'(x) = f(x)$ if $f$ is continuous. Direct application of FTC Part 2.
$G'(x) = f(x)$ if $f$ is continuous. Direct application of FTC Part 2.
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Find $\frac{d}{dx} \int_{2}^{x} (3t^2 + 2) , dt$.
Find $\frac{d}{dx} \int_{2}^{x} (3t^2 + 2) , dt$.
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$3x^2 + 2$. By FTC Part 2, derivative of integral equals the integrand.
$3x^2 + 2$. By FTC Part 2, derivative of integral equals the integrand.
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What does $\int_{a}^{a} f(x) , dx$ equal for any function $f$?
What does $\int_{a}^{a} f(x) , dx$ equal for any function $f$?
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- When integration limits are equal, the integral is zero.
- When integration limits are equal, the integral is zero.
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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 antiderivative of $f$, then $\int_{a}^{b} f(x) , dx = F(b) - F(a)$. This connects antiderivatives with definite integrals.
If $F$ is antiderivative of $f$, then $\int_{a}^{b} f(x) , dx = F(b) - F(a)$. This connects antiderivatives with definite integrals.
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State the antiderivative of $f(x) = \frac{1}{x}$.
State the antiderivative of $f(x) = \frac{1}{x}$.
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$F(x) = \ln |x| + C$. Standard antiderivative formula for reciprocal function.
$F(x) = \ln |x| + C$. Standard antiderivative formula for reciprocal function.
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