Applying Properties of Definite Integrals - AP Calculus BC
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Evaluate $\int_0^{\pi} \sin(x),dx$ using known values.
Evaluate $\int_0^{\pi} \sin(x),dx$ using known values.
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- $[-\cos(x)]_0^\pi = -(-1) - (-1) = 2$
- $[-\cos(x)]_0^\pi = -(-1) - (-1) = 2$
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What is the property of symmetry for even functions in definite integrals?
What is the property of symmetry for even functions in definite integrals?
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$\int_{-a}^a f(x),dx = 2\int_0^a f(x),dx$ if $f(x)$ is even. Even function symmetry doubles half-interval.
$\int_{-a}^a f(x),dx = 2\int_0^a f(x),dx$ if $f(x)$ is even. Even function symmetry doubles half-interval.
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What is $\frac{d}{dx} \int_a^x f(t),dt$?
What is $\frac{d}{dx} \int_a^x f(t),dt$?
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$f(x)$. Fundamental Theorem of Calculus, first part.
$f(x)$. Fundamental Theorem of Calculus, first part.
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Evaluate $\int_2^5 4,dx$ using properties.
Evaluate $\int_2^5 4,dx$ using properties.
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- $4(5-2) = 12$
- $4(5-2) = 12$
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Find the derivative: $\frac{d}{dx} \int_2^x \sin(t),dt$.
Find the derivative: $\frac{d}{dx} \int_2^x \sin(t),dt$.
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$\sin(x)$. By Fundamental Theorem of Calculus.
$\sin(x)$. By Fundamental Theorem of Calculus.
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Evaluate $\int_0^3 (x^2 - x),dx$ using linearity.
Evaluate $\int_0^3 (x^2 - x),dx$ using linearity.
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$\frac{9}{2}$. $\int_0^3 x^2 dx - \int_0^3 x dx = 9 - \frac{9}{2} = \frac{9}{2}$
$\frac{9}{2}$. $\int_0^3 x^2 dx - \int_0^3 x dx = 9 - \frac{9}{2} = \frac{9}{2}$
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What is the definite integral of a zero function over any interval?
What is the definite integral of a zero function over any interval?
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- Zero function integrates to zero.
- Zero function integrates to zero.
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Identify the integral property: $\int_a^b f(x),dx = -\int_b^a f(x),dx$.
Identify the integral property: $\int_a^b f(x),dx = -\int_b^a f(x),dx$.
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Reversal of Limits. Changes sign when limits are swapped.
Reversal of Limits. Changes sign when limits are swapped.
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Evaluate $\int_{-2}^2 x^3,dx$ using symmetry.
Evaluate $\int_{-2}^2 x^3,dx$ using symmetry.
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- $x^3$ is odd, so integral over symmetric limits is zero.
- $x^3$ is odd, so integral over symmetric limits is zero.
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What does the comparison property of definite integrals state?
What does the comparison property of definite integrals state?
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If $f(x) \leq g(x)$, then $\int_a^b f(x),dx \leq \int_a^b g(x),dx$. Preserves inequalities in integration.
If $f(x) \leq g(x)$, then $\int_a^b f(x),dx \leq \int_a^b g(x),dx$. Preserves inequalities in integration.
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What does the Mean Value Theorem for definite integrals state?
What does the Mean Value Theorem for definite integrals state?
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There exists $c \in [a, b]$ such that $f(c) = \frac{1}{b-a}\int_a^b f(x),dx$. Guarantees average value exists somewhere in interval.
There exists $c \in [a, b]$ such that $f(c) = \frac{1}{b-a}\int_a^b f(x),dx$. Guarantees average value exists somewhere in interval.
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What is the property of additivity for definite integrals?
What is the property of additivity for definite integrals?
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$\int_a^b f(x),dx + \int_b^c f(x),dx = \int_a^c f(x),dx$. Combines adjacent intervals into one integral.
$\int_a^b f(x),dx + \int_b^c f(x),dx = \int_a^c f(x),dx$. Combines adjacent intervals into one integral.
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How does the zero integral property affect definite integrals?
How does the zero integral property affect definite integrals?
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$\int_a^a f(x),dx = 0$. No area when upper and lower limits are equal.
$\int_a^a f(x),dx = 0$. No area when upper and lower limits are equal.
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Find $\int_{-3}^3 (x^4 + x^2),dx$ using symmetry.
Find $\int_{-3}^3 (x^4 + x^2),dx$ using symmetry.
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- Both functions are even, so use symmetry: $2\int_0^3 (x^4 + x^2)dx$
- Both functions are even, so use symmetry: $2\int_0^3 (x^4 + x^2)dx$
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What is $\frac{d}{dx} \int_0^x e^t,dt$?
What is $\frac{d}{dx} \int_0^x e^t,dt$?
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$e^x$. By Fundamental Theorem of Calculus.
$e^x$. By Fundamental Theorem of Calculus.
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What is a definite integral's value if $f(x) = 0$ for all $x$?
What is a definite integral's value if $f(x) = 0$ for all $x$?
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- Zero function has no area.
- Zero function has no area.
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What is the formula for the definite integral of a constant function $c$?
What is the formula for the definite integral of a constant function $c$?
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$\int_a^b c,dx = c(b-a)$. Constant times interval width.
$\int_a^b c,dx = c(b-a)$. Constant times interval width.
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What is the property of definite integrals when reversing limits?
What is the property of definite integrals when reversing limits?
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$\int_a^b f(x),dx = -\int_b^a f(x),dx$. Swapping limits changes sign.
$\int_a^b f(x),dx = -\int_b^a f(x),dx$. Swapping limits changes sign.
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Identify the property used: $\int_a^b (f(x) + g(x)),dx = \int_a^b f(x),dx + \int_a^b g(x),dx$.
Identify the property used: $\int_a^b (f(x) + g(x)),dx = \int_a^b f(x),dx + \int_a^b g(x),dx$.
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Linearity. Distributes integration over addition.
Linearity. Distributes integration over addition.
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Express the linearity property of definite integrals.
Express the linearity property of definite integrals.
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$\int_a^b (f(x) + g(x)),dx = \int_a^b f(x),dx + \int_a^b g(x),dx$. Integral of sum equals sum of integrals.
$\int_a^b (f(x) + g(x)),dx = \int_a^b f(x),dx + \int_a^b g(x),dx$. Integral of sum equals sum of integrals.
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Evaluate $\int_1^4 5x^2,dx$ using constant multiple property.
Evaluate $\int_1^4 5x^2,dx$ using constant multiple property.
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- $5 \cdot \int_1^4 x^2 dx = 5 \cdot 21 = 105$
- $5 \cdot \int_1^4 x^2 dx = 5 \cdot 21 = 105$
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What is the effect of integration limits on the definite integral?
What is the effect of integration limits on the definite integral?
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$\int_a^b f(x),dx$ depends only on $f(x)$ between $a$ and $b$. Values outside limits don't affect the integral.
$\int_a^b f(x),dx$ depends only on $f(x)$ between $a$ and $b$. Values outside limits don't affect the integral.
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Evaluate $\int_0^1 x^3,dx$ using known antiderivatives.
Evaluate $\int_0^1 x^3,dx$ using known antiderivatives.
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$\frac{1}{4}$. $\left[\frac{x^4}{4}\right]_0^1 = \frac{1}{4}$
$\frac{1}{4}$. $\left[\frac{x^4}{4}\right]_0^1 = \frac{1}{4}$
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Evaluate $\int_0^4 (x + 1),dx$ using properties.
Evaluate $\int_0^4 (x + 1),dx$ using properties.
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- $\int_0^4 x dx + \int_0^4 1 dx = 8 + 4 = 12$
- $\int_0^4 x dx + \int_0^4 1 dx = 8 + 4 = 12$
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State the property of definite integrals involving constant multiplication.
State the property of definite integrals involving constant multiplication.
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$\int_a^b c\cdot f(x),dx = c\cdot \int_a^b f(x),dx$. Constants factor out of integrals.
$\int_a^b c\cdot f(x),dx = c\cdot \int_a^b f(x),dx$. Constants factor out of integrals.
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Find the definite integral: $\int_0^2 3,dx$.
Find the definite integral: $\int_0^2 3,dx$.
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- $3(2-0) = 6$
- $3(2-0) = 6$
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Using properties, find $\int_0^3 (x^2 + 4x),dx$.
Using properties, find $\int_0^3 (x^2 + 4x),dx$.
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- $\int_0^3 x^2 dx + \int_0^3 4x dx = 9 + 18 = 27$
- $\int_0^3 x^2 dx + \int_0^3 4x dx = 9 + 18 = 27$
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Identify the property of definite integrals for an interval of zero length.
Identify the property of definite integrals for an interval of zero length.
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$\int_a^a f(x),dx = 0$. Zero width interval gives zero area.
$\int_a^a f(x),dx = 0$. Zero width interval gives zero area.
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What is the integral property that allows splitting at a point?
What is the integral property that allows splitting at a point?
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$\int_a^c f(x),dx = \int_a^b f(x),dx + \int_b^c f(x),dx$. Breaks integral at intermediate point.
$\int_a^c f(x),dx = \int_a^b f(x),dx + \int_b^c f(x),dx$. Breaks integral at intermediate point.
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What is the definite integral of an odd function over symmetric limits?
What is the definite integral of an odd function over symmetric limits?
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- Odd functions over symmetric intervals cancel.
- Odd functions over symmetric intervals cancel.
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