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AP Calculus AB : Interpretations and properties of definite integrals

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Example Question #5 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{-3}^{9}f(x)dx=-67\\&\int_{9}^{22}f(x)dx=74\\&\int_{37}^{22}f(x)dx=-18\\&\int_{39}^{37}f(x)dx=-67\\&\int_{42}^{39}f(x)dx=68\\&\int_{42}^{45}f(x)dx=86\\&\text{Calculate from the above values }\int_{-3}^{45}f(x)dx\end{align*}$

Possible Answers:

110

Correct answer:

110

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{-3}^{45}f(x)dx\\&\int_{-3}^{45}f(x)dx=\int_{-3}^{9}f(x)dx+\int_{9}^{22}f(x)dx-\int_{37}^{22}f(x)dx-\int_{39}^{37}f(x)dx-\int_{42}^{39}f(x)dx+\int_{42}^{45}f(x)dx\\&\int_{-3}^{45}f(x)dx=(-67)+(74)-(-18)-(-67)-(68)+(86)\\&\int_{-3}^{45}f(x)dx=110\\&\end{align*}$

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Example Question #6 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{-9}^{0}f(x)dx=62\\&\int_{13}^{0}f(x)dx=-46\\&\int_{18}^{13}f(x)dx=20\\&\text{Use the above values to find }\int_{-9}^{18}f(x)dx\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{-9}^{18}f(x)dx\\&\int_{-9}^{18}f(x)dx=\int_{-9}^{0}f(x)dx-\int_{13}^{0}f(x)dx-\int_{18}^{13}f(x)dx\\&\int_{-9}^{18}f(x)dx=(62)-(-46)-(20)\\&\int_{-9}^{18}f(x)dx=88\\&\end{align*}$

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Example Question #7 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{-10}^{3}f(x)dx=95\\&\int_{3}^{13}f(x)dx=63\\&\int_{23}^{13}f(x)dx=78\\&\int_{23}^{37}f(x)dx=-54\\&\int_{37}^{50}f(x)dx=-56\\&\int_{50}^{56}f(x)dx=77\\&\text{Use the above values to find }\int_{-10}^{56}f(x)dx\end{align*}$

Possible Answers:

203

172

Correct answer:

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{-10}^{56}f(x)dx\\&\int_{-10}^{56}f(x)dx=\int_{-10}^{3}f(x)dx+\int_{3}^{13}f(x)dx-\int_{23}^{13}f(x)dx+\int_{23}^{37}f(x)dx+\int_{37}^{50}f(x)dx+\int_{50}^{56}f(x)dx\\&\int_{-10}^{56}f(x)dx=(95)+(63)-(78)+(-54)+(-56)+(77)\\&\int_{-10}^{56}f(x)dx=47\\&\end{align*}$

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Example Question #8 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{-4}^{6}f(x)dx=5\\&\int_{12}^{6}f(x)dx=90\\&\int_{15}^{12}f(x)dx=73\\&\int_{19}^{15}f(x)dx=-70\\&\int_{22}^{19}f(x)dx=97\\&\int_{24}^{22}f(x)dx=36\\&\text{Calculate from the above values }\int_{-4}^{24}f(x)dx\end{align*}$

Possible Answers:

231

Correct answer:

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{-4}^{24}f(x)dx\\&\int_{-4}^{24}f(x)dx=\int_{-4}^{6}f(x)dx-\int_{12}^{6}f(x)dx-\int_{15}^{12}f(x)dx-\int_{19}^{15}f(x)dx-\int_{22}^{19}f(x)dx-\int_{24}^{22}f(x)dx\\&\int_{-4}^{24}f(x)dx=(5)-(90)-(73)-(-70)-(97)-(36)\\&\int_{-4}^{24}f(x)dx=-221\\&\end{align*}$

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Example Question #1 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{-4}^{0}f(x)dx=-13\\&\int_{15}^{0}f(x)dx=56\\&\int_{15}^{23}f(x)dx=-53\\&\int_{23}^{31}f(x)dx=96\\&\int_{31}^{32}f(x)dx=-44\\&\text{Calculate from the above values }\int_{-4}^{32}f(x)dx\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{-4}^{32}f(x)dx\\&\int_{-4}^{32}f(x)dx=\int_{-4}^{0}f(x)dx-\int_{15}^{0}f(x)dx+\int_{15}^{23}f(x)dx+\int_{23}^{31}f(x)dx+\int_{31}^{32}f(x)dx\\&\int_{-4}^{32}f(x)dx=(-13)-(56)+(-53)+(96)+(-44)\\&\int_{-4}^{32}f(x)dx=-70\\&\end{align*}$

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Example Question #10 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{3}^{6}f(x)dx=-8\\&\int_{6}^{16}f(x)dx=-87\\&\int_{24}^{16}f(x)dx=-85\\&\int_{39}^{24}f(x)dx=48\\&\int_{53}^{39}f(x)dx=42\\&\text{Determine }\int_{3}^{53}f(x)dx\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{3}^{53}f(x)dx\\&\int_{3}^{53}f(x)dx=\int_{3}^{6}f(x)dx+\int_{6}^{16}f(x)dx-\int_{24}^{16}f(x)dx-\int_{39}^{24}f(x)dx-\int_{53}^{39}f(x)dx\\&\int_{3}^{53}f(x)dx=(-8)+(-87)-(-85)-(48)-(42)\\&\int_{3}^{53}f(x)dx=-100\\&\end{align*}$

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Example Question #11 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{11}^{2}f(x)dx=-56\\&\int_{25}^{11}f(x)dx=-52\\&\int_{25}^{40}f(x)dx=-82\\&\int_{40}^{46}f(x)dx=-71\\&\int_{59}^{46}f(x)dx=-23\\&\text{Calculate from the above values }\int_{2}^{59}f(x)dx\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{2}^{59}f(x)dx\\&\int_{2}^{59}f(x)dx=-\int_{11}^{2}f(x)dx-\int_{25}^{11}f(x)dx+\int_{25}^{40}f(x)dx+\int_{40}^{46}f(x)dx-\int_{59}^{46}f(x)dx\\&\int_{2}^{59}f(x)dx=-(-56)-(-52)+(-82)+(-71)-(-23)\\&\int_{2}^{59}f(x)dx=-22\\&\end{align*}$

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Example Question #12 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{3}^{10}f(x)dx=38\\&\int_{25}^{10}f(x)dx=-14\\&\int_{30}^{25}f(x)dx=76\\&\int_{30}^{36}f(x)dx=27\\&\int_{36}^{45}f(x)dx=9\\&\text{Use the above values to find }\int_{3}^{45}f(x)dx\end{align*}$

Possible Answers:

136

Correct answer:

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{3}^{45}f(x)dx\\&\int_{3}^{45}f(x)dx=\int_{3}^{10}f(x)dx-\int_{25}^{10}f(x)dx-\int_{30}^{25}f(x)dx+\int_{30}^{36}f(x)dx+\int_{36}^{45}f(x)dx\\&\int_{3}^{45}f(x)dx=(38)-(-14)-(76)+(27)+(9)\\&\int_{3}^{45}f(x)dx=12\\&\end{align*}$

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Example Question #13 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{22}^{8}f(x)dx=-68\\&\int_{22}^{28}f(x)dx=68\\&\int_{28}^{29}f(x)dx=44\\&\text{Calculate from the above values }\int_{8}^{29}f(x)dx\end{align*}$

Possible Answers:

180

112

Correct answer:

180

Explanation:

$\begin{align*}&\text{One of the principles of integrals is one of summation. If the}\\&\text{values of an integral across certain sets of points is known,}\\&\text{this could potentially be used to find integral values at other}\\&\text{points. Imagine a set of points that are in ascending numerical}\\&\text{value: a, b, c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\\&\text{Another principle of integrals is that if an integral value}\\&\text{is known, to take the same integral backwards will give a negative}\\&\text{of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing this, we can calculate }\int_{8}^{29}f(x)dx\\&\int_{8}^{29}f(x)dx=-\int_{22}^{8}f(x)dx+\int_{22}^{28}f(x)dx+\int_{28}^{29}f(x)dx\\&\int_{8}^{29}f(x)dx=-(-68)+(68)+(44)\\&\int_{8}^{29}f(x)dx=180\\&\end{align*}$

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Example Question #14 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{13}^{4}f(x)dx=42\\&\int_{17}^{13}f(x)dx=-56\\&\int_{17}^{28}f(x)dx=35\\&\int_{28}^{33}f(x)dx=-99\\&\int_{4}^{48}f(x)dx=-10\\&\text{Determine }\int_{48}^{33}f(x)dx\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A principle of integrals is one of summation. If the values}\\&\text{of an integral across given sets of points is known, they could}\\&\text{ be used to find integral values across other points. Imagine}\\&\text{a set of points that are in ascending numerical value: a, b,}\\&\text{c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\text{ and likewise: }\int_a^d-\int_a^b-\int_b^c=\int_c^d\\&\text{Also, if an integral value is known, the same integral backwards}\\&\text{will give a negative of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing all this, we can calculate }\int_{48}^{33}f(x)dx\\&\int_{4}^{48}f(x)dx=-\int_{13}^{4}f(x)dx-\int_{17}^{13}f(x)dx+\int_{17}^{28}f(x)dx+\int_{28}^{33}f(x)dx-\int_{48}^{33}f(x)dx\\&-\int_{48}^{33}f(x)dx=\int_{4}^{48}f(x)dx+\int_{13}^{4}f(x)dx+\int_{17}^{13}f(x)dx-\int_{17}^{28}f(x)dx-\int_{28}^{33}f(x)dx\\&-\int_{48}^{33}f(x)dx=-10+(42)+(-56)-(35)-(-99)\\&\int_{48}^{33}f(x)dx=-40\\&\end{align*}$

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AP Calculus AB : Interpretations and properties of definite integrals

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #5 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #6 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #7 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #8 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #1 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #10 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #11 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #12 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #13 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #14 : Basic Properties Of Definite Integrals (Additivity And Linearity)

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