AP Calculus AB : Integrals

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

Example Question #12 : Interpretations And Properties Of Definite Integrals

\displaystyle \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:

\displaystyle 5.76

\displaystyle 31.12

\displaystyle 0.66

\displaystyle 12.68

Correct answer:

\displaystyle 5.76

Explanation:

\displaystyle \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*}

Example Question #13 : Interpretations And Properties Of Definite Integrals

\displaystyle \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:

\displaystyle -4.87

\displaystyle -2.12

\displaystyle -19.92

\displaystyle -0.35

Correct answer:

\displaystyle -2.12

Explanation:

\displaystyle \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*}

Example Question #14 : Interpretations And Properties Of Definite Integrals

\displaystyle \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:

\displaystyle 17.94

\displaystyle 37.07

\displaystyle 0.68

\displaystyle 5.98

Correct answer:

\displaystyle 5.98

Explanation:

\displaystyle \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*}

Example Question #15 : Interpretations And Properties Of Definite Integrals

\displaystyle \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:

\displaystyle -98.53

\displaystyle -24.63

\displaystyle -62.12

\displaystyle -10.71

Correct answer:

\displaystyle -10.71

Explanation:

\displaystyle \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*}

Example Question #16 : Interpretations And Properties Of Definite Integrals

\displaystyle \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:

\displaystyle 0.62

\displaystyle 0.18

\displaystyle 0.07

\displaystyle 3.16

Correct answer:

\displaystyle 0.62

Explanation:

\displaystyle \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*}

Example Question #17 : Interpretations And Properties Of Definite Integrals

\displaystyle \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:

\displaystyle 39.36

\displaystyle 1147.00

\displaystyle 370.00

\displaystyle 58.73

Correct answer:

\displaystyle 370.00

Explanation:

\displaystyle \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*}

Example Question #21 : Interpretations And Properties Of Definite Integrals

\displaystyle \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:

\displaystyle 1.85

\displaystyle 11.67

\displaystyle 0.19

\displaystyle 6.67

Correct answer:

\displaystyle 1.85

Explanation:

\displaystyle \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*}

Example Question #21 : Interpretations And Properties Of Definite Integrals

\displaystyle \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:

\displaystyle 5728.88

\displaystyle 186.67

\displaystyle 1736.02

\displaystyle 275.56

Correct answer:

\displaystyle 1736.02

Explanation:

\displaystyle \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*}

Example Question #23 : Interpretations And Properties Of Definite Integrals

A pot of water begins at a temperature of \displaystyle 20^\circ C and is heated at a rate of \displaystyle \tfrac{10}{(t+1)^2} degrees Celsius per minute. What will the temperature of the water be after 4 minutes?

Possible Answers:

\displaystyle 28^\circ C

\displaystyle 32^\circ C

\displaystyle 30.5^\circ C

\displaystyle 25^\circ C

\displaystyle 24^\circ C

Correct answer:

\displaystyle 28^\circ C

Explanation:

Let \displaystyle T(t) denote the temperature of the pot after \displaystyle t minutes.

 

The first thing to realize is that the quantity \displaystyle \frac{-10}{(t+1)^2} is the derivative of \displaystyle T(t). Using the fundamental theorem of Calculus, we know that 

\displaystyle \int_{0}^{4}\frac{10}{(t+1)^2} = T(4) - T(0)

 

From there, we just need to solve the integral. Letting u = t+1, du=dt, we have the following:

\displaystyle \int_{0}^{4}\frac{10}{(t+1)^2}dx = 10\int_{0}^{4} \frac{1}{u^2}du = \frac{-10}{(t+1)}]_0^4 = -2 + 10 = 8

Where the second equality follows by the power rule, and re-substituting t+1 = u.

Thus, we now have the equation \displaystyle 8 = T(4) - T(0). Because we know that the water started at \displaystyle 20^\circ C, all we need to do is rearrange and substitute.

\displaystyle 8 + T(0) = T(4)

\displaystyle 8 + 20 = T(4)

\displaystyle 28 = T(4)

 

Yielding our final answer, \displaystyle 28^\circ C

Example Question #61 : Integrals

A ball is thrown into the air. It's height, after t seconds is modeled by the formula:

h(t)=-15t^2+30t feet. 

At what time will the velocity equal zero?

Possible Answers:

1.5s

3s

1s

0s

5s

Correct answer:

1s

Explanation:

In order to find where the velocity is equal to zero, take the derivative of the function and set it equal to zero. 

h(t) = –15t+ 30t

h'(t) = –30t + 30

0 = –30t + 30

Then solve for "t".

–30 = –30t

t = 1

The velocity will be 0 at 1 second. 

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