AP Calculus AB: Riemann Sums and Definite Integral Notation — Drill 1

About This Drill

AP Calculus AB: Riemann Sums and Definite Integral Notation — Drill 1 is a Math practice drill covering Riemann Sums and Definite Integrals. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

Practice approximating definite integrals using left, right, midpoint, and trapezoidal Riemann sums, and interpret the definite integral as a limit of Riemann sums. These skills appear on both the multiple-choice and free-response sections of the AP Calculus AB exam.

Questions in This Drill

1. The table below shows selected values of a continuous function \( f \).
   
   

   x	0	2	4	6
   f(x)	1	5	3	9

   
   Using a right Riemann sum with 3 subintervals of equal width, which of the following approximates \( \int_0^6 f(x)\,dx \)?
2. Which of the following expressions is equivalent to \( \int_1^4 x^2\,dx \)?
3. A continuous function \( g \) is positive and strictly decreasing on \( [2,8] \). Which of the following must be true about Riemann sum approximations of \( \int_2^8 g(x)\,dx \)?
4. The table below shows selected values of a continuous function \( h \).
   
   

   x	0	3	6	9
   h(x)	2	8	4	6

   
   Using the trapezoidal rule with 3 subintervals of equal width, which of the following approximates \( \int_0^9 h(x)\,dx \)?
5. A particle moves along a straight line with velocity \( v(t) \) feet per second, where \( v(t) \) may be positive or negative. A student writes the expression \( \displaystyle\lim_{n \to \infty} \sum_{k=1}^{n} v\left(\dfrac{5k}{n}\right) \cdot \dfrac{5}{n} \) to model a physical quantity over \( [0,5] \). Which of the following best describes what this expression represents?