Exploring Riemann Sums and Integrals

What is the Riemann sum for f(x) = x^2 on [1, 3] using 4 subintervals with right endpoints?

• 8.5

• 9.25

• 12.0

• 10.75

Calculate the left Riemann sum for f(x) = sin(x) on [0, π] using 4 subintervals.

• (π/4) * (1 + √2)

• (π/2) * (1 + √2)

• (π/4) * (1 + 2√2)

• (π/4) * (1 - √2)

Estimate the area under the curve of f(x) = e^x from x = 0 to x = 1 using a midpoint Riemann sum with 2 subintervals.

• 0.5 * (e^0.25 + e^0.75)

• 0.5 * (e^0 + e^1)

• e^0.5

• 0.5 * (e^0.1 + e^0.9)

Using the definition of a definite integral, express the integral of f(x) = 1/x from 1 to 2 as a limit of Riemann sums.

• lim (n→∞) Σ (from i=1 to n) (1/(2 + i/n))(1/n)

• lim (n→∞) Σ (from i=1 to n) (1/(1 + i))(1/n)

• lim (n→∞) Σ (from i=1 to n) (1/(1 + i/n^2))(1/n)

• lim (n→∞) Σ (from i=1 to n) (1/(1 + i/n))(1/n)

Evaluate the limit of the sum as n approaches infinity for the Riemann sum of f(x) = 3x^2 on [0, 2].

• 4

• 2

• 8

• 6

Apply the Fundamental Theorem of Calculus to find the integral of f(x) = 2x from x = 1 to x = 4.

• 5

• 10

• 15

• 20

Interpret the graphical representation of the function f(x) = x^3 and identify the area under the curve from x = 0 to x = 2.

• 2

• 8

• 6

• 4

What is the trapezoidal approximation for the integral of f(x) = x^2 from x = 0 to x = 2 using 2 subintervals?

• 4

• 3

• 2

• 5

Determine the average value of the function f(x) = 4 - x^2 on the interval [0, 2].

• 2.6666666666666665

• 2.0

• 1.5

• 3.5

Explain how the Fundamental Theorem of Calculus connects differentiation and integration with an example.

• It asserts that the derivative of a function is always zero.

• The Fundamental Theorem of Calculus connects differentiation and integration by stating that the integral of a function can be computed using its antiderivative.

• The theorem only applies to polynomial functions and not to other types of functions.

• The Fundamental Theorem of Calculus states that differentiation and integration are unrelated processes.