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AP Calculus AB Practice Quiz: Approximating Areas with Riemann Sums

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 11 questions to check your progress.

Question 1 of 11

A Riemann sum is used to approximate the value of which of the following?

All Questions (11)

A Riemann sum is used to approximate the value of which of the following?

A) The derivative of a function at a point

B) A definite integral

C) An indefinite integral

D) The limit of a function

Correct Answer: B

A definite integral represents the net signed area under a curve. A Riemann sum approximates this area by summing the areas of geometric shapes, such as rectangles, over partitions of the interval. [cite: 2625]

The function `f` is continuous and selected values are given in the table below. Using three subintervals of equal width, what is the left Riemann sum approximation of the integral of `f(x)` from x=0 to x=6? | x | 0 | 2 | 4 | 6 | |---|---|---|---|---| | f(x)| 5 | 7 | 11| 17|

A) 46

B) 56

C) 69

D) 70

Correct Answer: A

The width of each of the three subintervals is (6-0)/3 = 2. A left Riemann sum uses the function value at the left endpoint of each subinterval. The approximation is `Δx * [f(0) + f(2) + f(4)] = 2 * [5 + 7 + 11] = 2 * 23 = 46`. [cite: 2626, 2627]

The graph of the function `f` is shown. What is the right Riemann sum approximation of the integral of `f(x)` from x=1 to x=5 using four subintervals of equal width?

A) 8

B) 10

C) 12

D) 14

Correct Answer: C

The interval [1, 5] is divided into four subintervals, so the width of each is (5-1)/4 = 1. The subintervals are [1, 2], [2, 3], [3, 4], and [4, 5]. A right Riemann sum uses the right endpoints: x=2, 3, 4, 5. From the graph, f(2)=2, f(3)=3, f(4)=4, and f(5)=3. The approximation is `1*f(2) + 1*f(3) + 1*f(4) + 1*f(5) = 1*2 + 1*3 + 1*4 + 1*3 = 12`. [cite: 2626, 2627]

A car's velocity `v(t)` is measured at various times `t` as shown in the table. Using a trapezoidal sum with the nonuniform subintervals given by the table, what is the approximate distance the car traveled from t=0 to t=10? | t (sec) | 0 | 3 | 8 | 10 | |---|---|---|---|---| | v(t) (ft/sec)| 10| 20| 18| 12|

A) 155 ft

B) 165 ft

C) 170 ft

D) 180 ft

Correct Answer: B

The distance is the integral of velocity. The trapezoidal sum is calculated for each subinterval. The subintervals have widths 3-0=3, 8-3=5, and 10-8=2. The approximation is `(1/2)(v(0)+v(3))*3 + (1/2)(v(3)+v(8))*5 + (1/2)(v(8)+v(10))*2 = (1/2)(10+20)*3 + (1/2)(20+18)*5 + (1/2)(18+12)*2 = (1/2)(30)*3 + (1/2)(38)*5 + (1/2)(30)*2 = 45 + 95 + 30 = 165` feet. [cite: 2627]

What is the midpoint Riemann sum approximation of the integral of `x^2` from x=0 to x=4 using two subintervals of equal width?

A) 10

B) 17

C) 20

D) 34

Correct Answer: C

With two subintervals on [0, 4], the width of each is (4-0)/2 = 2. The subintervals are [0, 2] and [2, 4]. The midpoints are x=1 and x=3. The function is f(x) = x^2. The midpoint Riemann sum approximation is `width * (f(midpoint1) + f(midpoint2)) = 2 * (f(1) + f(3)) = 2 * (1^2 + 3^2) = 2 * (1 + 9) = 2 * 10 = 20`. [cite: 2626, 2627]

If a function `f` is continuous and decreasing on the interval `[a, b]`, which of the following statements must be true about the left Riemann sum approximation of its definite integral from `a` to `b`?

A) It is an overestimate.

B) It is an underestimate.

C) It is exactly equal to the definite integral.

D) It is impossible to determine without knowing the concavity.

Correct Answer: A

For a decreasing function, the left endpoint of any subinterval is the highest point in that subinterval. Therefore, the rectangle formed using the left endpoint's height will have an area greater than the area under the curve for that subinterval, resulting in an overall overestimate of the definite integral. [cite: 2629]

A right Riemann sum is used to approximate the area under the curve of a function `g(x)`. If `g(x)` is a continuous and increasing function on the interval `[a, b]`, the approximation will be:

A) An overestimate.

B) An underestimate.

C) Exactly equal to the true area.

D) Dependent on the concavity of `g(x)`.

Correct Answer: A

For an increasing function, the right endpoint of any subinterval is the highest point in that subinterval. The rectangle formed using the right endpoint's height will have an area greater than the area under the curve for that subinterval, leading to an overall overestimate. [cite: 2629]

The trapezoidal sum approximation for the integral of `f(x)` from `a` to `b` is an overestimate of the true value. Which of the following statements about the function `f` on the interval `[a, b]` would explain this?

A) The function `f` is increasing.

B) The function `f` is decreasing.

C) The graph of `f` is concave up.

D) The graph of `f` is concave down.

Correct Answer: C

The top of each trapezoid in a trapezoidal sum is a secant line connecting two points on the curve. If the function is concave up, the curve lies below its secant lines. Therefore, the area of each trapezoid will be greater than the area under the curve on that subinterval, resulting in an overestimate. [cite: 2629]

Water is pumped into a tank at a rate modeled by `R(t)` gallons per minute, where `t` is measured in minutes. The definite integral of `R(t)` from t=0 to t=60 gives the total gallons of water pumped into the tank in the first hour. Which of the following expressions represents a right Riemann sum approximation of this integral with 6 subintervals of equal width?

A) 10 * [R(0) + R(10) + R(20) + R(30) + R(40) + R(50)]

B) 10 * [R(10) + R(20) + R(30) + R(40) + R(50) + R(60)]

C) 6 * [R(0) + R(10) + R(20) + R(30) + R(40) + R(50) + R(60)]

D) 6 * [R(10) + R(20) + R(30) + R(40) + R(50) + R(60)]

Correct Answer: B

The total interval is from 0 to 60 minutes. With 6 subintervals of equal width, the width of each subinterval is Δt = (60-0)/6 = 10 minutes. A right Riemann sum uses the right endpoint of each subinterval. The right endpoints are t = 10, 20, 30, 40, 50, and 60. The sum is the width of the subintervals (10) multiplied by the sum of the function values at these right endpoints. [cite: 2626, 2627]

Let `f` be a function that is positive, increasing, and concave down on the interval `[1, 5]`. Let `L`, `R`, and `T` be the left Riemann sum, right Riemann sum, and trapezoidal sum approximations, respectively, for the integral of `f(x)` from 1 to 5, each using the same number of subintervals. Let `I` be the exact value of the integral. Which of the following inequalities is true?

A) L < T < I < R

B) L < I < T < R

C) R < I < T < L

D) T < I < L < R

Correct Answer: A

Since `f` is increasing, the left sum (L) is an underestimate and the right sum (R) is an overestimate (`L < I < R`). Since `f` is concave down, the secant lines for the trapezoidal sum (T) lie below the curve, making T an underestimate (`T < I`). For any subinterval, the area of the trapezoid is greater than the area of the left rectangle because the function is increasing. Thus, `L < T`. Combining these inequalities gives the final order: `L < T < I < R`. [cite: 2629]

Which of the following is a primary reason for using a numerical method like a Riemann sum or a trapezoidal sum to evaluate a definite integral?

A) These methods always provide the exact value of the integral.

B) They are only applicable to functions represented by a table of values.

C) The antiderivative of the integrand cannot be found or is difficult to find in terms of elementary functions.

D) They are used to find the derivative of the function.

Correct Answer: C

The Fundamental Theorem of Calculus allows for exact evaluation of definite integrals, but it requires finding an antiderivative. For many functions, an elementary antiderivative does not exist or is very difficult to find. Numerical methods provide a way to approximate the value of the definite integral in such cases, or when the function is only defined by discrete data points. [cite: 2628]