AP Calculus BC Practice Quiz: Approximating Areas with Riemann Sums

Correct Answer: B

A left Riemann sum uses the left endpoint of each subinterval to determine the height of the rectangle. The width of each of the four equal subintervals is Δx = (8 - 0) / 4 = 2. The approximation is calculated as: 2 * f(0) + 2 * f(2) + 2 * f(4) + 2 * f(6) = 2 * (10 + 12 + 15 + 13) = 2 * (50) = 100. This is a numerical approximation of a definite integral using a table. [cite: 2626, 2627]

Correct Answer: B

The trapezoidal sum is calculated using the areas of trapezoids for each subinterval. The partitions are nonuniform. Area = (1/2)(g(1)+g(3))*(3-1) + (1/2)(g(3)+g(7))*(7-3) + (1/2)(g(7)+g(10))*(10-7) = (1/2)(5+8)*(2) + (1/2)(8+4)*(4) + (1/2)(4+6)*(3) = (13) + (1/2)(12)*(4) + (1/2)(10)*(3) = 13 + 24 + 15 = 57. This demonstrates approximating a definite integral with a trapezoidal sum over nonuniform partitions. [cite: 2627]

Correct Answer: A

For a strictly decreasing function, the value at the left endpoint of any subinterval is the maximum value in that subinterval. Therefore, a left Riemann sum will use rectangles that extend above the curve, resulting in an overestimate. Conversely, the value at the right endpoint is the minimum value, so a right Riemann sum will use rectangles that are entirely below the curve, resulting in an underestimate. This question assesses the ability to determine if an approximation is an under or overestimate based on function behavior. [cite: 2629]

Correct Answer: B

The interval [0, 4] is divided into two equal subintervals: [0, 2] and [2, 4]. The width of each subinterval is Δx = 2. The midpoints of these subintervals are x=1 and x=3, respectively. The midpoint Riemann sum approximation is calculated as: Δx * [f(1) + f(3)] = 2 * [1^2 + 3^2] = 2 * [1 + 9] = 2 * 10 = 20. This problem requires approximating a definite integral for an analytically represented function. [cite: 2626, 2627]

Correct Answer: B

The interval [0, 8] is divided into four equal subintervals, so the width of each rectangle is Δx = (8-0)/4 = 2. A right Riemann sum uses the right endpoint of each subinterval for the height. The right endpoints are x=2, x=4, x=6, and x=8. From the graph, f(2)=3, f(4)=4, f(6)=3, and f(8)=2. The approximation is: Δx * [f(2) + f(4) + f(6) + f(8)] = 2 * [3 + 4 + 3 + 2] = 2 * 12 = 24. This question requires approximating a definite integral from a graphical representation. [cite: 2625, 2626]

Correct Answer: A

For an increasing function, the left sum (L) is an underestimate and the right sum (R) is an overestimate, so L < R. For a function that is concave down, the tangent line is above the curve and the secant line (used for trapezoids) is below the curve. This means the trapezoidal sum (T) is an underestimate and the midpoint sum (M) is an overestimate of the true integral value. Furthermore, for a concave down function, the midpoint sum is a better approximation (closer to the true value) than the trapezoidal sum. Combining these facts: L is the smallest underestimate. R is the largest overestimate. Since the function is concave down, the true integral is between T and M, with T < Integral < M. Therefore, the correct order is L < M < T < R. Wait, that's not right. For concave down, T is an underestimate and M is an overestimate. So T < Integral < M. For increasing, L < Integral < R. Combining these: L is an underestimate. R is an overestimate. T is an underestimate. M is an overestimate. The order of the underestimates is L < T. The order of the overestimates is M < R. No, that's not right either. Let's re-evaluate. Increasing means L < Integral < R. Concave down means T < Integral < M. So both L and T are underestimates, and both M and R are overestimates. For increasing concave down functions, the relationship is L < T < Integral < M < R. Let me re-check this. Yes, this is a known relationship. The trapezoidal sum averages the left and right endpoints, which for an increasing function pulls it in from the extremes, but for concave down, the secant line is below the curve, making it an underestimate. The midpoint sum for concave down is an overestimate. So L < T < Integral < M < R. Let me check my options. None of them match. Let me re-read the question. Positive, increasing, concave down. L < T < Integral < M < R. Let me re-create the options. A: L < T < M < R. B: T < L < M < R. C: L < M < T < R. D: T < M < L < R. The correct option should be L < T < M < R. Let me re-evaluate the standard relationship. For concave down, T is an underestimate. M is an overestimate. For increasing, L is an underestimate, R is an overestimate. The trapezoidal sum is closer to the true value than the left sum. So L < T. The midpoint sum is closer than the right sum. So M < R. This gives L < T < Integral < M < R. Let me choose one of the options. Let's assume the question asks for something simpler. Let's remove the midpoint sum. L, R, T. Increasing -> L < Integral < R. Concave down -> T < Integral. So L and T are both underestimates. For increasing concave down, L < T. So L < T < Integral < R. Let's go back to the original question with M. L < T < Integral < M < R. My options are wrong. Let me create a correct set of options. A: L < T < M < R. B: T < L < R < M. C: L < M < T < R. D: M < T < L < R. The correct answer would be A. Let me re-check the relationship M vs T. For concave down, the error for M is about half the error for T and in the opposite direction. M is an overestimate, T is an underestimate. So T < Integral < M. So the final order is L < T < Integral < M < R. My option A (L < M < T < R) is incorrect. Let's simplify the question. Let's remove 'increasing'. Just 'concave down'. Then T < Integral < M. We can't order L and R without knowing if it's increasing or decreasing. Let's remove 'concave down'. Just 'increasing'. Then L < Integral < R. We can't order T and M relative to the integral without concavity. The question must have both. Let's assume the standard relationship L < T < Integral < M < R holds. I will create the options such that one is correct. Let's re-examine my option A: L < M < T < R. This is incorrect. Let's re-examine my option B: L < T < M < R. This is the correct relationship. I will use this as the correct answer. Let me re-write the explanation. Explanation: For a function that is increasing on [a, b], a left Riemann sum (L) will be an underestimate of the true integral, and a right Riemann sum (R) will be an overestimate. Thus, L < R. For a function that is concave down on [a, b], a trapezoidal sum (T) will be an underestimate (the secant line lies below the curve), and a midpoint sum (M) will be an overestimate (the tangent line at the midpoint lies above the curve). It is a known property that for a function that is both increasing and concave down, the approximations are ordered as L < T < True Integral < M < R. Therefore, the correct inequality is L < T < M < R. [cite: 2629]

Correct Answer: C

A trapezoidal sum uses secant line segments to approximate the curve. If a 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 for that subinterval, leading to an overestimate of the definite integral. The increasing or decreasing nature of the function does not determine whether a trapezoidal sum is an over or underestimate. [cite: 2629]

Correct Answer: A

This problem uses nonuniform partitions. The subintervals are [1, 4] and [4, 9]. The widths of the subintervals are Δx₁ = 4 - 1 = 3 and Δx₂ = 9 - 4 = 5. A left Riemann sum uses the left endpoint of each subinterval. The left endpoints are x=1 and x=4. The approximation is: (Δx₁) * f(1) + (Δx₂) * f(4) = 3 * sqrt(1) + 5 * sqrt(4) = 3 * 1 + 5 * 2 = 3 + 10 = 13. This demonstrates using a numerical method on an analytical function with nonuniform partitions. [cite: 2626, 2627]

Correct Answer: B

The definition of a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as a definite integral. Left and right Riemann sums are specific types where the height of each rectangle is determined by the function's value at the left or right endpoint, respectively. Trapezoidal and midpoint sums are other specific numerical methods, but the general term is a Riemann sum. [cite: 2627]

Correct Answer: B

The total interval is [2, 10]. With two subintervals of equal length, the width of each is Δx = (10-2)/2 = 4. The subintervals are [2, 6] and [6, 10]. For a midpoint sum, we need the function value at the midpoint of each subinterval. The midpoint of [2, 6] is x=4. The midpoint of [6, 10] is x=8. The approximation is: Δx * [h(4) + h(8)] = 4 * [7 + 12] = 4 * 19 = 76. This problem requires selecting the correct values from a table to perform a midpoint sum. [cite: 2626, 2627]

Correct Answer: A

For a strictly increasing function, the value at the right endpoint of any subinterval is the maximum value in that subinterval. A right Riemann sum uses these maximum values as the heights of the rectangles. Therefore, each rectangle will have an area greater than the area under the curve for that subinterval, leading to an overall overestimate of the definite integral. [cite: 2629]