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Chapter 4: Q. 41 (page 340)

In Exercises 39鈥�44, write out the sigma notation for the Riemann sum described in such a way that the only letter which appears in the general term of the sum is k. Don鈥檛 calculate the value of the sum; just write it down in sigma notation.
$f (x) = e^{x}, [a, b] = [1, 4], m i d p o i n t s u m, n = 6$

Short Answer

Expert verified

The midpoint sum is $\frac{1}{2} {鈭�}_{k = 1}^{6} e^{\frac{2 k + 3}{4}}$

Step by step solution

01

Step 1. Given Information

$f (x) = e^{x}, [a, b] = [1, 4], m i d p o i n t s u m, n = 6$

02

Step 1. Finding ∆x,xk and xk-1

$鈭� x = \frac{b - a}{n} = \frac{4 - 1}{n} = \frac{3}{6} = \frac{1}{2}$

$x_{k} = a + k 鈭� x = 1 + \frac{k}{2} = \frac{k + 2}{2} x_{k - 1} = a + (k - 1) 鈭� x = \frac{k - 1 + 2}{2} = \frac{k + 1}{2}$

and $\frac{x_{k - 1} + x_{k}}{2} = \frac{\frac{k + 1}{2} + \frac{k + 2}{2}}{2} = \frac{k + 1 + k + 2}{4} = \frac{2 k + 3}{4}$

03

Step 3. Finding Reimann Sum

We have $f (x) = e^{x}$ then $f (x_{k}) = e^{x_{k}}$ and $f (x_{k - 1}) = e^{x_{k - 1}}$

Now, ${鈭�}_{k = 1}^{n} f (\frac{x_{k - 1} + x_{k}}{2}) 鈭� x = {鈭�}_{k = 1}^{n} f (\frac{2 k + 3}{4}) \frac{1}{2} = \frac{1}{2} {鈭�}_{k = 1}^{n} e^{\frac{2 k + 3}{4}}$

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Most popular questions from this chapter

Repeat Exercise 13 for the function f shown above at the right, on the interval $[- 2, 2]$

Determine which of the limit of sums in Exercises 47鈥�52 are infinite and which are finite. For each limit of sums that is finite, compute its value

$\lim_{n 鈫� 鈭�} {鈭�}_{k = 1}^{n} (k^{2} + k + 1)$

What is the difference between an antiderivative of a function and the indefinite integral of a function?

Verify that $鈭� \cot x d x = \ln (\sin x) + C$ . (Do not try to solve the integral from scratch.)

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