AP Calculus AB : Integrals

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

Example Question #5 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

What is the domain of f(x)=\frac{x+5}{\sqrt{x^2-9}}?

Possible Answers:

(-\infty,-3]\cup [3,+\infty)

(-\infty,-3)\cup (3,+\infty)

(3,+\infty)

(-5,+\infty)

Correct answer:

(-\infty,-3)\cup (3,+\infty)

Explanation:

{\sqrt{x^2-9}}>0 because the denominator cannot be zero and square roots cannot be taken of negative numbers

x^2-9>0

x^2>9

\sqrt{x^2}>\sqrt{9}

\left | x \right |>3

x>3\: or\, x<-3

Example Question #6 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

If y-6x-x^{2}=4,

then at , what is 's instantaneous rate of change?

Possible Answers:

Correct answer:

Explanation:

The answer is 8.

 

y-6x-x^{2}=4

y=x^{2}+6x+4

y'=2x+6

Example Question #1 : Interpretations And Properties Of Definite Integrals

Which of the following represents the graph of the polar function  in Cartestian coordinates?

Possible Answers:

Correct answer:

Explanation:

First, mulitply both sides by r. 

Then, use the identities  and .

The answer is .

Example Question #1 : Interpretations And Properties Of Definite Integrals

What is the average value of the function  from  to ?

Possible Answers:

Correct answer:

Explanation:

The average function value is given by the following formula:

, evaluated from  to .

Example Question #2 : Interpretations And Properties Of Definite Integrals

h(x)=\frac{g(x)}{(1+f(x))}

If 

then find .

Possible Answers:

\frac{5}{2}

\frac{-1}{2}

Correct answer:

Explanation:

We see the answer is 0 after we do the quotient rule. 

 

h(x)=\frac{g(x)}{(1+f(x))}

h'(x)=\frac{g'(x)(1+f(x))-g(x)(f'(x))}{(1+f(x))^{2}}

Example Question #51 : Integrals

If g(x)=\int_{0}^{x^2}f(t)dt, then which of the following is equal to g'(x)?

Possible Answers:

f(x^{2})-f(0)

f(x^{2})

f(2x)

2xf(x^{2})

g(f(x^{2}))

Correct answer:

2xf(x^{2})

Explanation:

According to the Fundamental Theorem of Calculus, if we take the derivative of the integral of a function, the result is the original function. This is because differentiation and integration are inverse operations.

For example, if h(x)=\int_{a}^{x}f(u)du, where  is a constant, then h'(x)=f(x).

We will apply the same principle to this problem. Because the integral is evaluated from 0 to x^{2}, we must apply the chain rule.

g'(x)=\frac{d}{dx}\int_{0}^{x^{2}}f(t)dt=f(x^{2})\cdot \frac{d}{dx}(x^{2})

=2xf(x^{2})

The answer is 2xf(x^{2}).

Example Question #9 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

Possible Answers:

Correct answer:

Explanation:

Example Question #10 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

Possible Answers:

Correct answer:

Explanation:

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

Possible Answers:

Correct answer:

Explanation:

Example Question #12 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

Possible Answers:

Correct answer:

Explanation:

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