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AP Calculus AB : Basic properties of definite integrals (additivity and linearity)

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Example Question #21 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{-4}^{-3}f(x)dx=66\\&\int_{-3}^{4}f(x)dx=-32\\&\int_{6}^{14}f(x)dx=25\\&\int_{-4}^{14}f(x)dx=3\\&\text{Calculate from the above values }\int_{6}^{4}f(x)dx\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A principle of integrals is one of summation. If the values}\\&\text{of an integral across given sets of points is known, they could}\\&\text{ be used to find integral values across other points. Imagine}\\&\text{a set of points that are in ascending numerical value: a, b,}\\&\text{c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\text{ and likewise: }\int_a^d-\int_a^b-\int_b^c=\int_c^d\\&\text{Also, if an integral value is known, the same integral backwards}\\&\text{will give a negative of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing all this, we can calculate }\int_{6}^{4}f(x)dx\\&\int_{-4}^{14}f(x)dx=\int_{-4}^{-3}f(x)dx+\int_{-3}^{4}f(x)dx-\int_{6}^{4}f(x)dx+\int_{6}^{14}f(x)dx\\&-\int_{6}^{4}f(x)dx=\int_{-4}^{14}f(x)dx-\int_{-4}^{-3}f(x)dx-\int_{-3}^{4}f(x)dx-\int_{6}^{14}f(x)dx\\&-\int_{6}^{4}f(x)dx=3-(66)-(-32)-(25)\\&\int_{6}^{4}f(x)dx=56\\&\end{align*}$

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Example Question #22 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{10}^{19}f(x)dx=-40\\&\int_{19}^{30}f(x)dx=1\\&\int_{42}^{30}f(x)dx=-12\\&\int_{4}^{42}f(x)dx=-19\\&\text{Use the above values to find }\int_{10}^{4}f(x)dx\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A principle of integrals is one of summation. If the values}\\&\text{of an integral across given sets of points is known, they could}\\&\text{ be used to find integral values across other points. Imagine}\\&\text{a set of points that are in ascending numerical value: a, b,}\\&\text{c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\text{ and likewise: }\int_a^d-\int_a^b-\int_b^c=\int_c^d\\&\text{Also, if an integral value is known, the same integral backwards}\\&\text{will give a negative of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing all this, we can calculate }\int_{10}^{4}f(x)dx\\&\int_{4}^{42}f(x)dx=-\int_{10}^{4}f(x)dx+\int_{10}^{19}f(x)dx+\int_{19}^{30}f(x)dx-\int_{42}^{30}f(x)dx\\&-\int_{10}^{4}f(x)dx=\int_{4}^{42}f(x)dx-\int_{10}^{19}f(x)dx-\int_{19}^{30}f(x)dx+\int_{42}^{30}f(x)dx\\&-\int_{10}^{4}f(x)dx=-19-(-40)-(1)+(-12)\\&\int_{10}^{4}f(x)dx=-8\\&\end{align*}$

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Example Question #23 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{-4}^{1}f(x)dx=-81\\&\int_{14}^{10}f(x)dx=80\\&\int_{14}^{24}f(x)dx=-4\\&\int_{24}^{31}f(x)dx=96\\&\int_{-4}^{31}f(x)dx=-11\\&\text{Use the above values to find }\int_{10}^{1}f(x)dx\end{align*}$

Possible Answers:

102

Correct answer:

Explanation:

$\begin{align*}&\text{A principle of integrals is one of summation. If the values}\\&\text{of an integral across given sets of points is known, they could}\\&\text{ be used to find integral values across other points. Imagine}\\&\text{a set of points that are in ascending numerical value: a, b,}\\&\text{c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\text{ and likewise: }\int_a^d-\int_a^b-\int_b^c=\int_c^d\\&\text{Also, if an integral value is known, the same integral backwards}\\&\text{will give a negative of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing all this, we can calculate }\int_{10}^{1}f(x)dx\\&\int_{-4}^{31}f(x)dx=\int_{-4}^{1}f(x)dx-\int_{10}^{1}f(x)dx-\int_{14}^{10}f(x)dx+\int_{14}^{24}f(x)dx+\int_{24}^{31}f(x)dx\\&-\int_{10}^{1}f(x)dx=\int_{-4}^{31}f(x)dx-\int_{-4}^{1}f(x)dx+\int_{14}^{10}f(x)dx-\int_{14}^{24}f(x)dx-\int_{24}^{31}f(x)dx\\&-\int_{10}^{1}f(x)dx=-11-(-81)+(80)-(-4)-(96)\\&\int_{10}^{1}f(x)dx=-58\\&\end{align*}$

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Example Question #24 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{0}^{4}f(x)dx=67\\&\int_{5}^{4}f(x)dx=-6\\&\int_{8}^{22}f(x)dx=72\\&\int_{22}^{23}f(x)dx=-54\\&\int_{0}^{23}f(x)dx=96\\&\text{Calculate from the above values }\int_{8}^{5}f(x)dx\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A principle of integrals is one of summation. If the values}\\&\text{of an integral across given sets of points is known, they could}\\&\text{ be used to find integral values across other points. Imagine}\\&\text{a set of points that are in ascending numerical value: a, b,}\\&\text{c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\text{ and likewise: }\int_a^d-\int_a^b-\int_b^c=\int_c^d\\&\text{Also, if an integral value is known, the same integral backwards}\\&\text{will give a negative of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing all this, we can calculate }\int_{8}^{5}f(x)dx\\&\int_{0}^{23}f(x)dx=\int_{0}^{4}f(x)dx-\int_{5}^{4}f(x)dx-\int_{8}^{5}f(x)dx+\int_{8}^{22}f(x)dx+\int_{22}^{23}f(x)dx\\&-\int_{8}^{5}f(x)dx=\int_{0}^{23}f(x)dx-\int_{0}^{4}f(x)dx+\int_{5}^{4}f(x)dx-\int_{8}^{22}f(x)dx-\int_{22}^{23}f(x)dx\\&-\int_{8}^{5}f(x)dx=96-(67)+(-6)-(72)-(-54)\\&\int_{8}^{5}f(x)dx=-5\\&\end{align*}$

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Example Question #21 : Basic Properties Of Definite Integrals (Additivity And Linearity)

$\begin{align*}&\int_{1}^{7}f(x)dx=16\\&\int_{13}^{28}f(x)dx=-82\\&\int_{1}^{28}f(x)dx=-30\\&\text{Use the above values to find }\int_{13}^{7}f(x)dx\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A principle of integrals is one of summation. If the values}\\&\text{of an integral across given sets of points is known, they could}\\&\text{ be used to find integral values across other points. Imagine}\\&\text{a set of points that are in ascending numerical value: a, b,}\\&\text{c, and d. It follows that:}\\&\int_a^d=\int_a^b+\int_b^c+\int_c^d\text{ and likewise: }\int_a^d-\int_a^b-\int_b^c=\int_c^d\\&\text{Also, if an integral value is known, the same integral backwards}\\&\text{will give a negative of the original value:}\\&\int_a^b=-\int_b^a\\&\text{Knowing all this, we can calculate }\int_{13}^{7}f(x)dx\\&\int_{1}^{28}f(x)dx=\int_{1}^{7}f(x)dx-\int_{13}^{7}f(x)dx+\int_{13}^{28}f(x)dx\\&-\int_{13}^{7}f(x)dx=\int_{1}^{28}f(x)dx-\int_{1}^{7}f(x)dx-\int_{13}^{28}f(x)dx\\&-\int_{13}^{7}f(x)dx=-30-(16)-(-82)\\&\int_{13}^{7}f(x)dx=-36\\&\end{align*}$

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Example Question #51 : Interpretations And Properties Of Definite Integrals

Given that $\int_{5}^{1}(2u)dx = 8$ and $\int_{3}^{2}(-u)dx = 4$ , find the value of the following expression:

$\int_{1}^{3}(u)dx - \int_{5}^{4}(u)dx + \int_{2}^{4}(u)dx$

Possible Answers:

Correct answer:

Explanation:

First, simplifying the given's gives us

$\int_{5}^{1}(2u)dx = -2\int_{1}^{5}(u)dx = 8$

$\int_{1}^{5}(u)dx = -4$

And

$\int_{3}^{2}(-u)dx = \int_{2}^{3}(u)dx = 4$

Our goal is to get the given expression into terms of these two integrals. Our first step will be to try and get a $\int_{1}^{5}(u)dx$ from our expression.

First note,

$-\int_{5}^{4}(u)dx = \int_{4}^{5}(u)dx$

And for the third term,

$\int_{2}^{4}(u)dx = \int_{2}^{3}(u)dx + \int_{3}^{4}(u)dx$

Putting these facts together, we can rewrite the original expression as

$\int_{1}^{3}(u)dx +(\int_{4}^{5}(u)dx) + (\int_{2}^{3}(u)dx + \int_{3}^{4}(u)dx)$

Rearranging,

$(\int_{1}^{3}(u)dx + \int_{3}^{4}(u)dx + \int_{4}^{5}(u)dx) + \int_{2}^{3}(u)dx$

The three terms in parentheses can all be brought together, as the top limit of the previous integral matches the bottom limit of the next integral. Thus, we now have

$\int_{1}^{5}(u)dx + \int_{2}^{3}(u)dx$

Substituting in our given's, this simplifies to -4 + 4 = 0

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Example Question #21 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Evaluate the definite integral

$\int_{\pi}^{0}4(3x^2-sin(x)+2)dx$

Possible Answers:

$12\pi^2$

$-12\pi^2$

$4\pi^3+8\pi-8$

$-4\pi^3-8\pi+8$

Correct answer:

$-4\pi^3-8\pi+8$

Explanation:

Here we are using several basic properties of definite integrals as well as the fundamental theorem of calculus.

First, you can pull coefficients out to the front of integrals.

$\int_{\pi}^{0}4(3x^2-sin(x)+2)dx=4\int_{\pi}^{0}(3x^2-sin(x)+2)dx$

Second, we notice that our lower bound is bigger than our upper bound. You can switch the upper and lower bounds if you also switch the sign.

$4\int_{\pi}^{0}(3x^2-sin(x)+2)dx=-4\int_{0}^{\pi}(3x^2-sin(x)+2)dx$

Lastly, our integral "distributes" over addition and subtraction. That means you can split the integral by each term and integrate each term separately.

$-4\int_{0}^{\pi}(3x^2-sin(x)+2)dx=-4(\int_{0}^{\pi}3x^2dx-\int_{0}^{\pi}sin(x)dx+\int_{0}^{\pi}2dx)$

Now we integrate and calculate using the Fundamental Theorem of Calculus.

$-4(x^3+cos(x)+2x)]_{0}^{\pi}=-4[(\pi^3+cos(\pi)+2\pi)-(0^3+cos(0)+2(0))]$

$=-4[(\pi^3-1+2\pi)-1]=-4\pi^3-8\pi+8$

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Example Question #52 : Interpretations And Properties Of Definite Integrals

Solve:

$\int_{0}^{2} x dx+\int_{0}^{2}x^2dx$

Possible Answers:

$\frac{2}{3}$

None of the other answers

$\frac{14}{3}$

$\frac{8}{3}$

Correct answer:

$\frac{14}{3}$

Explanation:

Rather than solve each integral independently, we can use the property of linearity to add the integrals together:

$\int_{0}^{2} x dx+\int_{0}^{2}x^2dx = \int_{0}^{2}(x+x^2)dx=\frac{x^2}{2}+\frac{x^3}{3}\left.\begin{matrix} 2\\0 \end{matrix}\right|$

The integrals were solved using the following rule:

$\int x^n dx=\frac{x^{n+1}}{n+1}+C$

Finally, we evaluate the definite integral by evaluating the answer at the upper bound and subtracting the answer evaluated at the lower bound:

$\frac{8}{3}+2-(0+0)=\frac{14}{3}$

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AP Calculus AB : Basic properties of definite integrals (additivity and linearity)

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #21 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #22 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #23 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #24 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #21 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #51 : Interpretations And Properties Of Definite Integrals

Example Question #21 : Basic Properties Of Definite Integrals (Additivity And Linearity)

Example Question #52 : Interpretations And Properties Of Definite Integrals

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