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AP Calculus AB 6.3 Riemann Sums Summation Notation and Definite Integral Notation Questions

Connect Riemann sums with definite integrals by interpreting limiting sums and representing integrals with summation notation.

Syllabus
Effective Fall 2025
Course
AP Calculus AB

Exam points

  • interpret a Riemann sum as accumulated area or a definite integral
  • translate between summation notation, limits and definite-integral notation

Covered syllabus points

LIM-5.B—Interpret the limiting case of the Riemann sum as a definite integralLIM-5.C—Represent the limiting case of the Riemann sum as a definite integral

Question 1

[Maximum number: 1]

For a certain continuous function f, the right Riemann sum approximation of ∫02f(x)dx\int_{0}^{2} f(x) d x with n subintervals of equal length is 2(n+1)(3n+2)n2\frac{2(n+1)(3 n+2)}{n^{2}} for all n. What is the value of ∫02f(x)dx\int_{0}^{2} f(x) d x ?

Graph of \(f\)

Graph of \(f\)

A

2

B

6

C

12

D

20

Question 2

[Maximum number: 1]

Choose the integral that is the limit of the Riemann Sum: nit n approaches to infinity of sum from k equals 1 to n of left parenthesis left parenthesis start fraction numerator 3 k over denominator n end fraction plus 2 right parenthesis squared end exponent times left parenthesis 3 over n right parenthesis right parenthesis .

A

(A) ategral from 2 to 5 of left parenthesis x plus 2 right parenthesis squared end exponent d x

B

(B) ategral from 0 to 3 of left parenthesis 3 x plus 2 right parenthesis squared end exponent d x

C

(C) ategral from 2 to 5 of x squared end exponent dx

D

(D) ategral from 0 to 3 of x squared end exponent dx

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