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Learning Outcomes

1. Use sigma notation to represent summation.
2. Use the sum of rectangular areas to approximate the area under a curve.
3. Understand the notation of Â Riemann Sums.

Textbook

• Chapter 5.1 Â Approximating Areas Â Â

Textbook Assignment

• p. 523: Â  Â 2, 12, 14-17 all

WeBWork Assignment

• Integration-Riemann Sums

Exit problemsÂ of the sessionÂ

1. Compute $L_4$ and $R_4$ to approximate the area under the curve $f(x)=x^2+1$ on Â $[0, 4]$ . Explain whether it is an overestimation or underestimation.Â

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Key Concepts

• Sigma notation of the form $\displaystyle\sum_{i=1}^{n}a_i$ is useful to express long sums of values in compact form.Â
• Riemann Sums are expressions of the form $\displaystyle\sum_{i=1}^{n}f(x^*_i)\Delta x$, and can be used to estimate the area under the curve $y=f(x)$.
• Approximate the area under the curve $y=f(x)$ on the interval Â $[a, b]$ :
1. Divide the interval Â $[a, b]$ Â into Â $n$ Â subintervals with equal length $\Delta x=\dfrac{b-a}{n}$;
2. Let $x_i=a+i\cdot\Delta x$;
3. We can either use Left Endpoint Approximation: Â

$A\approx L_n=f(x_0)\Delta x+f(x_1)\Delta x+\cdots+f(x_{n-1})\Delta x=\displaystyle\sum_{i=1}^{n}f(x_{i-1})\Delta x$

or Right Endpoint Approximation:Â

$A\approx R_n=f(x_1)\Delta x+f(x_2)\Delta x+\cdots+f(x_{n})\Delta x=\displaystyle\sum_{i=1}^{n}f(x_i)\Delta x$

#### Videos and Practice Problems of Selected Topics

1. $\rhd$ Summation notation and sums (4:26) Write $1+\cdots + 10$ andÂ $1+\cdots + 100$ using summation notation. Â Expand the sum $\displaystyle\sum_{i=0}^{50}\pi i^2$.
2. * Summation notation A review with multiple-choice questions.
3. $\rhd$ Worked examples: summation notation (5:20)
• Consider the sum $2+5+8+11$. Find the expression that is equal to this sum.
• Expand $\displaystyle\sum_{n=1}^{4}\dfrac{k}{n+1}$.
4. * Practice: Summation notation. (4 problems)
5. $\rhd$ Riemann approximation introduction (6:44) Approximate the area under the curve $y=x^2+1$ using rectangles.
6. $\rhd$ Left and right Riemann sums (4:00) Given the graph of a function $y=g(x)$, find the left and right Riemann sums that approximate the area under $y=g(x)$ between $x=2$ and $x=4$.
7. * Practice: Over and under estimation of Riemann sums. (4 problems)

#### STEM Application

MAT1475 Optimization Assignment