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Quiz: Riemann Sums and the Fundamental Theorem

Test your understanding of Riemann sums and the FTC with these review questions.

1. A Riemann sum approximates:

  1. The derivative of a function
  2. The area under a curve using rectangles
  3. The maximum of a function
  4. The limit of a sequence
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The correct answer is B. A Riemann sum approximates the area under a curve by dividing the region into rectangles and summing their areas. More rectangles give better approximations.

Concept Tested: Riemann Sum

2. In a left Riemann sum, the height of each rectangle is:

  1. The function value at the right endpoint
  2. The function value at the left endpoint
  3. The average of left and right values
  4. The function value at the midpoint
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The correct answer is B. In a left Riemann sum, each rectangle's height is the function value at the left endpoint of the subinterval.

Concept Tested: Left Riemann Sum

3. The definite integral ∫ₐᵇ f(x) dx is defined as:

  1. F(b) + F(a)
  2. The limit of Riemann sums as n→∞
  3. f(b) − f(a)
  4. The derivative of F(x)
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The correct answer is B. The definite integral is defined as the limit of Riemann sums as the number of subintervals approaches infinity (and width approaches zero).

Concept Tested: Definite Integral

4. The Fundamental Theorem of Calculus, Part 1 states that d/dx[∫ₐˣ f(t) dt] equals:

  1. f(a)
  2. f(x)
  3. F(x) − F(a)
  4. ∫ₐˣ f'(t) dt
Show Answer

The correct answer is B. FTC Part 1: The derivative of an accumulation function is the integrand evaluated at the upper limit. d/dx[∫ₐˣ f(t) dt] = f(x).

Concept Tested: FTC Part One

5. The Fundamental Theorem of Calculus, Part 2 states that ∫ₐᵇ f(x) dx equals:

  1. f(b) − f(a)
  2. F(b) − F(a), where F' = f
  3. F(a) − F(b)
  4. f(b) + f(a)
Show Answer

The correct answer is B. FTC Part 2 (Evaluation Theorem): If F is an antiderivative of f, then ∫ₐᵇ f(x) dx = F(b) − F(a). This connects antiderivatives to definite integrals.

Concept Tested: FTC Part Two

6. What is ∫₀² x² dx?

  1. 4
  2. 8/3
  3. 4/3
  4. 2
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The correct answer is B. F(x) = x³/3. By FTC: ∫₀² x² dx = F(2) − F(0) = 8/3 − 0 = 8/3.

Concept Tested: Evaluation Theorem

7. An accumulation function F(x) = ∫ₐˣ f(t) dt represents:

  1. The slope of f at x
  2. The accumulated area under f from a to x
  3. The maximum of f
  4. The average value of f
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The correct answer is B. The accumulation function represents the signed area under f from a to x. As x increases, it "accumulates" more area.

Concept Tested: Accumulation Function

8. If d/dx[∫₀ˣ² f(t) dt], you must use:

  1. FTC Part 1 only
  2. FTC Part 2 only
  3. FTC Part 1 combined with Chain Rule
  4. Integration by parts
Show Answer

The correct answer is C. When the upper limit is a function of x (like x²), apply FTC Part 1 and then the Chain Rule: d/dx[∫₀^(g(x)) f(t) dt] = f(g(x)) · g'(x).

Concept Tested: FTC Chain Rule

9. The net signed area can be:

  1. Only positive
  2. Only negative
  3. Positive, negative, or zero
  4. Only zero
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The correct answer is C. Net signed area counts area above the x-axis as positive and below as negative. The definite integral gives net signed area, which can be positive, negative, or zero.

Concept Tested: Net Signed Area

10. The trapezoidal rule approximates area using:

  1. Rectangles with left endpoints
  2. Rectangles with right endpoints
  3. Trapezoids connecting function values
  4. Triangles
Show Answer

The correct answer is C. The trapezoidal rule uses trapezoids instead of rectangles, connecting consecutive function values with line segments. It's generally more accurate than left or right sums.

Concept Tested: Trapezoidal Rule