Riemann Sums & Definite Integrals (College Board AP® Calculus AB)

Exam Questions

34 mins17 questions

3 marks

$t$

(seconds)

100

120

$f (t)$

(liters per second)

0.008

0.012

0.008

0.004

A customer is filling up a bottle of olive oil from a large container in a health food shop. The rate of flow of the olive oil is modeled by a differentiable function $f$ , where $f (t)$ is measured in liters per second and $t$ is measured in seconds since filling the bottle began. Selected values of $f (t)$ are given in the table.

Using correct units, interpret the meaning of $\int_{30}^{100} f (t) d t$ in the context of the problem. Use a right Riemann sum with the three subintervals [30, 60], [60, 80], and [80, 100] to approximate the value of $\int_{30}^{100} f (t) d t$

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1 mark

$x$	0	1	3	6
$f (x)$	12	10	7	11
$f^{'} (x)$	-1	-2	3	1

$f$ is a differentiable function. The table shown gives values of the function $f$ and its first derivative at selected values of $x$ .

Let $g$ be the function defined by $g (x) = 2 x^{3} + \int_{0}^{x} f^{'} (t) d t$ . Find $g (3)$ . Show the work that leads to your answer.

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3 marks

Is the function $g$ defined in part (a) increasing, decreasing, or neither at $x = 3$ ? Justify your answer.

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2 marks

$t$ (minutes)	0	3	7	12	15
$M (t)$ (degrees Fahrenheit)	212.0	165.8	125.9	97.3	87.0

The temperature of coffee in a mug at time $t$ is modeled by a strictly decreasing, twice-differentiable function $M$ , where $M (t)$ is measured in degrees Fahrenheit and $t$ is measured in minutes. At time $t = 0$ , the temperature of the coffee is $212 ° F$ . The mug of coffee is then left to cool, beginning at time $t = 0$ . Values of $M (t)$ at selected times $t$ for the first 15 minutes are given in the table above.

Use the data in the table to evaluate $\int_{0}^{15} M^{'} (t) d t$ . Using correct units, interpret the meaning of $\int_{0}^{15} M^{'} (t) d t$ in the context of this problem.

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3 marks

For $0 \leq t \leq 15$ , the average temperature of the water in the mug is $\frac{1}{15} \int_{0}^{15} M (t) d t$ . Use a left Riemann sum with the four subintervals indicated by the data in the table to approximate $\frac{1}{15} \int_{0}^{15} M (t) d t$ . Does this approximation overestimate or underestimate the average temperature of the coffee over these 15 minutes? Explain your reasoning.

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2 marks

For $15 \leq t \leq 20$ , the function $M$ that models the coffee temperature has first derivative given by $M^{'} (t) = - 18.45 e^{- 0.125 t}$ . Based on the model, what is the temperature of the coffee at time $t = 20$ ?

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2 marks

$r$ (meters)	0	0.5	1	2	3
$f (r)$ (kilograms per square meter)	2	5	9	12	14

$r$

(meters)

0.5

$f (r)$

(kilograms per square meter)

The density of algae on the surface of a circular garden pond at a distance of $r$ meters from the center of the pond is given by an increasing, differentiable function $f$ , where $f (r)$ is measured in kilograms per square meter. Values of $f (r)$ for selected values of $r$ are given in the table above.

The total mass, in kilograms, of algae in the pond is given by the integral expression $2 π \int_{0}^{3} r f (r) d r$ . Approximate the value of $2 π \int_{0}^{3} r f (r) d r$ using a right Riemann sum with the four subintervals indicated by the data in the table.

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2 marks

Is the approximation found in part (a) an overestimate or underestimate of the total mass of algae in the pond? Explain your reasoning.

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