Riemann Sums, Summation Notation, Definite Integral Notation Practice

Sample questions

5 of 5 — sign in to practice the rest with adaptive difficulty and mastery tracking.

1. Sample 1difficulty 2/5

   $\displaystyle\int_a^b f(x)\,dx$ represents

   • A

     Average of $f(a)$ and $f(b)$

   • B

     Total area between the graph and $x$-axis

   • C

     Length of the curve

   • D

     Net <strong>signed</strong> area (above axis positive, below axis negative)

     check_circle

   Why

   Definite integral: signed area.

   View full solution & statsarrow_forward

2. Sample 2difficulty 3/5

   $\displaystyle\lim_{n\to\infty}\sum_{i=1}^n \dfrac{1}{n}\left( \dfrac{i}{n}\right)^2 =$

   • A

     $\tfrac{1}{4}$

   • B

     $\tfrac{1}{2}$

   • C

     $\tfrac{1}{3}$

     check_circle
   • D

     $1$

   Why

   Right-Riemann sum for $\int_0^1 x^2\,dx = 1/3$.

   View full solution & statsarrow_forward

3. Sample 3difficulty 3/5

   $\displaystyle\lim_{n\to\infty}\sum_{i=1}^n \dfrac{1}{n}\cdot e^{i/n}$ equals

   • A

     $1$

   • B

     $0$

   • C

     $e - 1$

     check_circle
   • D

     $e$

   Why

   Right Riemann sum on $[0, 1]$ for $f(x) = e^x$. Limit $= \int_0^1 e^x\,dx = e - 1$.

   View full solution & statsarrow_forward

4. Sample 4difficulty 3/5

   The areas above and below the $x$-axis are 5 and 3 respectively. Then $\int f(x)\,dx$ over the full interval equals:

   • A

     $15$

   • B

     $8$

   • C

     $-2$

   • D

     $2$

     check_circle

   Why

   Net signed area $= 5 - 3 = 2$.

   View full solution & statsarrow_forward

5. Sample 5difficulty 3/5

   The Riemann sum $\sum_{i=1}^n \dfrac{2}{n}\bigl[1 + (2 i/n)\bigr]$ is a sum approximating

   • A

     $\int_0^2 (1 + 2x)\,dx$

   • B

     $\int_0^2 (1 + x)\,dx$

     check_circle
   • C

     $\int_1^2 x\,dx$

   • D

     $\int_0^1 (1 + 2x)\,dx$

   Why

   Width $2/n$, points $0 + 2i/n$ on $[0, 2]$. Function $1 + x$.

   View full solution & statsarrow_forward