Calculus 2 Quiz: Riemann Sum Approximations

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Riemann Sum ApproximationsQuestion 1 of 5

A function $f(x)$ is continuous on $[2, 10]$ . Using a right Riemann sum with $n = 4$ subintervals, the approximation yields 84. Using a left Riemann sum with $n = 4$ subintervals, the approximation yields 76. If $f(x)$ is strictly increasing on $[2, 10]$ , what is the best estimate for the actual value of $\int_2^{10} f(x) \, dx$ ?

The integral equals exactly 80 since the function is strictly increasing

The integral is between 76 and 84, with 80 being a reasonable estimate

The integral is less than 76 since both approximations overestimate the true value

The integral is greater than 84 since both approximations underestimate the true value

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Calculus 2 Quiz

Calculus 2 Quiz: Riemann Sum Approximations

Practice Riemann Sum Approximations in Calculus 2 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Riemann Sum Approximations, giving you a quick way to practice the rules, question types, and explanations that matter most for Calculus 2.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The integral equals exactly 80 since the function is strictly increasing
The integral is between 76 and 84, with 80 being a reasonable estimate (correct answer)
The integral is less than 76 since both approximations overestimate the true value
The integral is greater than 84 since both approximations underestimate the true value

Explanation: For a strictly increasing function, the left Riemann sum underestimates the integral (gives 76) and the right Riemann sum overestimates it (gives 84). Therefore, the true value lies between 76 and 84. The average of these bounds, 80, serves as a reasonable estimate. Choice A incorrectly assumes the average is exact. Choice C incorrectly claims both sums overestimate. Choice D incorrectly claims both sums underestimate.

Question 2

Consider approximating $\int_0^3 \sqrt{9-x^2} \, dx$ using a midpoint Riemann sum with $n = 6$ subintervals. Which statement best describes the relationship between this approximation and the exact value?

The midpoint sum overestimates because $\sqrt{9-x^2}$ is concave down on $[0,3]$ , and midpoint sums overestimate concave down functions (correct answer)
The midpoint sum underestimates because $\sqrt{9-x^2}$ is concave down on $[0,3]$ , and midpoint sums underestimate concave down functions
The midpoint sum overestimates because $\sqrt{9-x^2}$ is concave up on $[0,3]$ , and midpoint sums overestimate concave up functions
The midpoint sum underestimates because $\sqrt{9-x^2}$ is concave up on $[0,3]$ , and midpoint sums underestimate concave up functions

Explanation: The function

f(x) = \sqrt{9-x^2}

has

f''(x) = -\frac{9}{(9-x^2)^{3/2}} < 0

(0,3)

, so it's concave down. For concave down functions, the midpoint sum overestimates the integral because the midpoint of each chord lies above the curve. Choice B has the correct concavity but wrong direction. Choices C and D incorrectly identify the concavity as upward.

Question 3

A continuous function $w(t)$ represents the rate of water flow into a tank (in gallons per minute) over a 12-minute period. Using a left Riemann sum with $\Delta t = 3$ minutes, the total volume is approximated as $3[w(0) + w(3) + w(6) + w(9)]$ . If this gives 84 gallons, and using $\Delta t = 2$ minutes gives $2[w(0) + w(2) + w(4) + w(6) + w(8) + w(10)] = 78$ gallons, what conclusion is most reasonable?

The actual volume is exactly 81 gallons since this is the average of the two approximations
The flow rate $w(t)$ is decreasing over the 12-minute period since the finer approximation gives a smaller result
The flow rate $w(t)$ is increasing over the 12-minute period since the finer approximation gives a smaller result (correct answer)
More information about $w(t)$ is needed to determine whether the flow rate is increasing or decreasing

Explanation: For left Riemann sums, if the function is increasing, using smaller intervals (finer approximation) gives a larger sum, approaching the true value from below. If the function is decreasing, smaller intervals give smaller sums. Since the

\Delta t = 2

approximation (78) is smaller than the

\Delta t = 3

approximation (84), this suggests

w(t)

is increasing. Choice A incorrectly assumes the average is exact. Choice B has the trend backwards. Choice D is unnecessarily cautious given the clear pattern.

Question 4

Consider the integral $\int_0^2 e^{-x^2} \, dx$ . A student uses both the midpoint rule and trapezoidal rule with $n = 4$ subintervals to approximate this integral. Which statement about the relative accuracy of these approximations is most likely correct?

The trapezoidal rule is more accurate because it uses more function evaluations per subinterval than the midpoint rule
The midpoint rule is more accurate because $e^{-x^2}$ is concave down on $[0,2]$ , and the midpoint rule has smaller error for concave down functions
Both methods have similar accuracy since they are both first-order approximation methods with the same number of subintervals
The midpoint rule is more accurate because it generally has error that is half that of the trapezoidal rule for smooth functions (correct answer)

Explanation: For smooth functions, the midpoint rule typically has error approximately half that of the trapezoidal rule with the same number of subintervals. This is a general theoretical result: the midpoint rule error is

O(h^2)

with a smaller constant than the trapezoidal rule. Choice A is wrong because both use the same number of evaluations. Choice B incorrectly analyzes concavity (

e^{-x^2}

changes concavity). Choice C is wrong because both are second-order methods, but with different error constants.

Question 5

A student calculates a left Riemann sum for $\int_0^4 h(x) \, dx$ using $n = 8$ subintervals and obtains 32. If the same integral is approximated using a left Riemann sum with $n = 4$ subintervals, and $h(x)$ is linear on each subinterval $[0,1]$ , $[1,2]$ , $[2,3]$ , and $[3,4]$ , what can be concluded about the $n = 4$ approximation?

The $n = 4$ approximation equals 16 since halving the intervals halves the sum for linear functions
The $n = 4$ approximation equals 32 since left sums are exact for piecewise linear functions when intervals align with linear pieces (correct answer)
The $n = 4$ approximation is less than 32 but the exact value cannot be determined without more information
The $n = 4$ approximation is greater than 32 since using fewer rectangles typically overestimates the integral

Explanation: When using left Riemann sums with subintervals that align exactly with the linear pieces of a piecewise linear function, the approximation becomes exact. This is because on each linear piece

[k, k+1]

, the left endpoint value

h(k)

multiplied by the interval length 1 gives exactly the area under that linear segment. Therefore, both the

n=4

and

n=8

approximations equal 32. Choice A incorrectly applies a scaling relationship. Choices C and D incorrectly assume the approximation has error when it's actually exact.