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AP Calculus AB : Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval

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Example Questions

Example Question #11 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

$\begin{align*}&\text{The flow of water into a tank is given as:}\\&\frac{dV(t)}{dt}=\frac{(79e^{(-t)})}{9}\\&\text{Find the change in volume over the time interval }t=[0.5,1.5]\end{align*}$

Possible Answers:

0.50

10.10

31.97

3.37

Correct answer:

3.37

Explanation:

$\begin{align*}&\text{If the function defining a rate of change is known, the total}\\&\text{change over a given interval of time is only a matter of integrating}\\&\text{this rate function over this same time interval. Considering}\\&\text{the function given:}\\&\text{Utilizing integral rules:}\\&\int[e^{ax}]=\frac{e^{ax}}{a}\\&\int_{0.5}^{1.5}(\frac{(79e^{(-t)})}{9})dt=-\frac{(79e^{(-t)})}{9}|_{0.5}^{1.5}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{0.5}^{1.5}(\frac{(79e^{(-t)})}{9})dt=-\frac{(79e^{(-(1.5))})}{9}-(-\frac{(79e^{(-(0.5))})}{9})\\&\int_{0.5}^{1.5}(\frac{(79e^{(-t)})}{9})dt=3.37\end{align*}$

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Example Question #11 : Interpretations And Properties Of Definite Integrals

$\begin{align*}&\text{The flow of water into a tank is given as:}\\&\frac{dV(t)}{dt}=\frac{(7\cdot 3^{(\frac{t}{3})})}{2}\\&\text{Find the change in volume over the time interval }t=[0.5,2]\end{align*}$

Possible Answers:

68.06

8.40

1.25

19.33

Correct answer:

8.40

Explanation:

$\begin{align*}&\text{If the function defining a rate of change is known, the total}\\&\text{change over a given interval of time is only a matter of integrating}\\&\text{this rate function over this same time interval. Considering}\\&\text{the function given:}\\&\text{Utilizing integral rules:}\\&\int[b^{ax}]=\frac{b^{ax}}{aln(b)}\\&\int_{0.5}^{2}(\frac{(7\cdot 3^{(\frac{t}{3})})}{2})dt=\frac{(21\cdot 3^{(\frac{t}{3})})}{(2ln(3))}|_{0.5}^{2}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{0.5}^{2}(\frac{(7\cdot 3^{(\frac{t}{3})})}{2})dt=\frac{(21\cdot 3^{(\frac{(2)}{3})})}{(2ln(3))}-(\frac{(21\cdot 3^{(\frac{(0.5)}{3})})}{(2ln(3))})\\&\int_{0.5}^{2}(\frac{(7\cdot 3^{(\frac{t}{3})})}{2})dt=8.40\end{align*}$

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Example Question #12 : Interpretations And Properties Of Definite Integrals

$\begin{align*}&\text{The rate of change of the length of an elastic is:}\\&\frac{dl(t)}{dt}=\frac{(17sin(t + 2))}{4}\\&\text{Determine the change in length over the time interval }t=[5,6.5]\end{align*}$

Possible Answers:

5.76

31.12

0.66

12.68

Correct answer:

5.76

Explanation:

$\begin{align*}&\text{If the function defining a rate of change is known, the total}\\&\text{change over a given interval of time is only a matter of integrating}\\&\text{this rate function over this same time interval. Considering}\\&\text{the function given:}\\&\text{Utilizing integral rules:}\\&\int[sin(ax)]=-\frac{cos(ax)}{a}\\&\int_{5}^{6.5}(\frac{(17sin(t + 2))}{4})dt=-\frac{(17cos(t + 2))}{4}|_{5}^{6.5}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{5}^{6.5}(\frac{(17sin(t + 2))}{4})dt=-\frac{(17cos((6.5) + 2))}{4}-(-\frac{(17cos((5) + 2))}{4})\\&\int_{5}^{6.5}(\frac{(17sin(t + 2))}{4})dt=5.76\end{align*}$

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Example Question #13 : Interpretations And Properties Of Definite Integrals

$\begin{align*}&\text{The flow of water into a tank is given as:}\\&\frac{dV(t)}{dt}=\frac{(57cos(3t))}{8}\\&\text{Find the change in volume over the time interval }t=[1,6]\end{align*}$

Possible Answers:

-4.87

-2.12

-19.92

-0.35

Correct answer:

-2.12

Explanation:

$\begin{align*}&\text{If the function defining a rate of change is known, the total}\\&\text{change over a given interval of time is only a matter of integrating}\\&\text{this rate function over this same time interval. Considering}\\&\text{the function given:}\\&\text{Utilizing integral rules:}\\&\int[cos(ax)]=\frac{sin(ax)}{a}\\&\int_{1}^{6}(\frac{(57cos(3t))}{8})dt=\frac{(19sin(3t))}{8}|_{1}^{6}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{1}^{6}(\frac{(57cos(3t))}{8})dt=\frac{(19sin(3(6)))}{8}-(\frac{(19sin(3(1)))}{8})\\&\int_{1}^{6}(\frac{(57cos(3t))}{8})dt=-2.12\end{align*}$

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Example Question #14 : Interpretations And Properties Of Definite Integrals

$\begin{align*}&\text{The velocity of a particle is given as:}\\&v(t)=\frac{(12e^{(t)})}{37}\\&\text{Find its change in position over the time interval }t=[0.5,3]\end{align*}$

Possible Answers:

17.94

37.07

0.68

5.98

Correct answer:

5.98

Explanation:

$\begin{align*}&\text{If the function defining a rate of change is known, the total}\\&\text{change over a given interval of time is only a matter of integrating}\\&\text{this rate function over this same time interval. Considering}\\&\text{the function given:}\\&\text{Utilizing integral rules:}\\&\int[e^{ax}]=\frac{e^{ax}}{a}\\&\int_{0.5}^{3}(\frac{(12e^{(t)})}{37})dt=\frac{(12e^{(t)})}{37}|_{0.5}^{3}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{0.5}^{3}(\frac{(12e^{(t)})}{37})dt=\frac{(12e^{((3))})}{37}-(\frac{(12e^{((0.5))})}{37})\\&\int_{0.5}^{3}(\frac{(12e^{(t)})}{37})dt=5.98\end{align*}$

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Example Question #15 : Interpretations And Properties Of Definite Integrals

$\begin{align*}&\text{The rate of change of the length of an elastic is:}\\&\frac{dl(t)}{dt}=\frac{(73sin(t + 1))}{11}\\&\text{Determine the change in length over the time interval }t=[3,5]\end{align*}$

Possible Answers:

-98.53

-24.63

-62.12

-10.71

Correct answer:

-10.71

Explanation:

$\begin{align*}&\text{If the function defining a rate of change is known, the total}\\&\text{change over a given interval of time is only a matter of integrating}\\&\text{this rate function over this same time interval. Considering}\\&\text{the function given:}\\&\text{Utilizing integral rules:}\\&\int[sin(ax)]=-\frac{cos(ax)}{a}\\&\int_{3}^{5}(\frac{(73sin(t + 1))}{11})dt=-\frac{(73cos(t + 1))}{11}|_{3}^{5}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{3}^{5}(\frac{(73sin(t + 1))}{11})dt=-\frac{(73cos((5) + 1))}{11}-(-\frac{(73cos((3) + 1))}{11})\\&\int_{3}^{5}(\frac{(73sin(t + 1))}{11})dt=-10.71\end{align*}$

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Example Question #16 : Interpretations And Properties Of Definite Integrals

$\begin{align*}&\text{The rate of change of the length of an elastic is:}\\&\frac{dl(t)}{dt}=\frac{(71sin(3t))}{7}\\&\text{Determine the change in length over the time interval }t=[4,6]\end{align*}$

Possible Answers:

0.62

0.18

0.07

3.16

Correct answer:

0.62

Explanation:

$\begin{align*}&\text{If the function defining a rate of change is known, the total}\\&\text{change over a given interval of time is only a matter of integrating}\\&\text{this rate function over this same time interval. Considering}\\&\text{the function given:}\\&\text{Utilizing integral rules:}\\&\int[sin(ax)]=-\frac{cos(ax)}{a}\\&\int_{4}^{6}(\frac{(71sin(3t))}{7})dt=-\frac{(71cos(3t))}{21}|_{4}^{6}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{4}^{6}(\frac{(71sin(3t))}{7})dt=-\frac{(71cos(3(6)))}{21}-(-\frac{(71cos(3(4)))}{21})\\&\int_{4}^{6}(\frac{(71sin(3t))}{7})dt=0.62\end{align*}$

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Example Question #17 : Interpretations And Properties Of Definite Integrals

$\begin{align*}&\text{The rate of change of the length of an elastic is:}\\&\frac{dl(t)}{dt}=\frac{(37t)}{2}\\&\text{Determine the change in length over the time interval }t=[3,7]\end{align*}$

Possible Answers:

39.36

1147.00

370.00

58.73

Correct answer:

370.00

Explanation:

$\begin{align*}&\text{If the function defining a rate of change is known, the total}\\&\text{change over a given interval of time is only a matter of integrating}\\&\text{this rate function over this same time interval. Considering}\\&\text{the function given:}\\&\text{Utilizing integral rules:}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\int_{3}^{7}(\frac{(37t)}{2})dt=\frac{(37t^{2})}{4}|_{3}^{7}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{3}^{7}(\frac{(37t)}{2})dt=\frac{(37(7)^{2})}{4}-(\frac{(37(3)^{2})}{4})\\&\int_{3}^{7}(\frac{(37t)}{2})dt=370.00\end{align*}$

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Example Question #21 : Interpretations And Properties Of Definite Integrals

$\begin{align*}&\text{The velocity of a particle is given as:}\\&v(t)=\frac{9}{17}\\&\text{Find its change in position over the time interval }t=[3,6.5]\end{align*}$

Possible Answers:

1.85

11.67

0.19

6.67

Correct answer:

1.85

Explanation:

$\begin{align*}&\text{If the function defining a rate of change is known, the total}\\&\text{change over a given interval of time is only a matter of integrating}\\&\text{this rate function over this same time interval. Considering}\\&\text{the function given:}\\&\text{Utilizing integral rules:}\\&\int x^{a}=\frac{x^{a+1}}{a+1}\\&\int_{3}^{6.5}(\frac{9}{17})dt=\frac{(9t)}{17}|_{3}^{6.5}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{3}^{6.5}(\frac{9}{17})dt=\frac{(9(6.5))}{17}-(\frac{(9(3))}{17})\\&\int_{3}^{6.5}(\frac{9}{17})dt=1.85\end{align*}$

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Example Question #21 : Interpretations And Properties Of Definite Integrals

$\begin{align*}&\text{The rate of change of the length of an elastic is:}\\&\frac{dl(t)}{dt}=\frac{(50e^{(t)})}{7}\\&\text{Determine the change in length over the time interval }t=[0.5,5.5]\end{align*}$

Possible Answers:

5728.88

186.67

1736.02

275.56

Correct answer:

1736.02

Explanation:

$\begin{align*}&\text{If the function defining a rate of change is known, the total}\\&\text{change over a given interval of time is only a matter of integrating}\\&\text{this rate function over this same time interval. Considering}\\&\text{the function given:}\\&\text{Utilizing integral rules:}\\&\int[e^{ax}]=\frac{e^{ax}}{a}\\&\int_{0.5}^{5.5}(\frac{(50e^{(t)})}{7})dt=\frac{(50e^{(t)})}{7}|_{0.5}^{5.5}\\&\text{Now we just subtract the function value at the lower bound from that of the upper bound:}\\&\int_{0.5}^{5.5}(\frac{(50e^{(t)})}{7})dt=\frac{(50e^{((5.5))})}{7}-(\frac{(50e^{((0.5))})}{7})\\&\int_{0.5}^{5.5}(\frac{(50e^{(t)})}{7})dt=1736.02\end{align*}$

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AP Calculus AB : Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #11 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

Example Question #11 : Interpretations And Properties Of Definite Integrals

Example Question #12 : Interpretations And Properties Of Definite Integrals

Example Question #13 : Interpretations And Properties Of Definite Integrals

Example Question #14 : Interpretations And Properties Of Definite Integrals

Example Question #15 : Interpretations And Properties Of Definite Integrals

Example Question #16 : Interpretations And Properties Of Definite Integrals

Example Question #17 : Interpretations And Properties Of Definite Integrals

Example Question #21 : Interpretations And Properties Of Definite Integrals

Example Question #21 : Interpretations And Properties Of Definite Integrals

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