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Chapter 4: Q. 43 (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.
$f (x) = \sin x, [a, b] = [0, 蟺], t r a p e z o i d s u m, n = 4$

Short Answer

Expert verified

The trapezoid sum is $\frac{蟺}{4} {鈭�}_{k = 1}^{4} \frac{\sin (\frac{蟺 (k + 1)}{4}) + \sin (\frac{蟺办}{4})}{2}$

Step by step solution

01

Step 1. Given Information

We have $f (x) = \sin x, [a, b] = [0, 蟺], t r a p e z o i d s u m, n = 4$

02

Step 2. Finding ∆x,xk and xk+1

$鈭� x = \frac{b - a}{n} = \frac{蟺 - 0}{n} = \frac{蟺}{4}$

$x_{k} = a + k 鈭� x = \frac{蟺办}{4} x_{k + 1} = a + (k + 1) 鈭� x = \frac{蟺 (k + 1)}{4}$

03

Step 3. Finding Riemann Sums

We have $f (x) = \sin x$ then $f (x_{k + 1}) = f (\frac{蟺 (k + 1)}{4}) = \sin (\frac{蟺 (k + 1)}{4}) f (x_{k}) = f (\frac{蟺办}{4}) = \sin (\frac{蟺办}{4})$

and role="math" localid="1649354205627" $\frac{f (x_{k + 1}) + f (x_{k})}{2} = \frac{\sin (\frac{蟺 (k + 1)}{4}) + \sin (\frac{蟺办}{4})}{2}$

Now, Riemann Sum

role="math" localid="1649354244791"

= {鈭�}_{k = 1}^{n} \frac{f (x_{k + 1}) + f (x_{k})}{2} 鈭� x = \frac{蟺}{4} {鈭�}_{k = 1}^{4} \frac{\sin (\frac{蟺 (k + 1)}{4}) + \sin (\frac{蟺办}{4})}{2}

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Most popular questions from this chapter

Show by exhibiting a counterexample that, in general, $鈭� f (x) g (x) d x 鈮� (鈭� f (x) d x) (鈭� g (x) d x)$ . In other words, find two functions f and g so that the integral of their product is not equal to the product of their integrals.

Find the sum or quantity without completely expanding or calculating any sums.

Given ${鈭�}_{k = 3}^{10} a_{k} = 12$ and ${鈭�}_{k = 2}^{10} a_{k} = 23$ , find $a_{2}$ .

Without using absolute values, how many definite integrals would we need in order to calculate the absolute area between f(x) = sin x and the x-axis on $[- \frac{蟺}{2}, 2 蟺]$ ?

Will the absolute area be positive or negative, and why? Will the signed area will be positive or negative, and why?

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.

$鈭� (x^{2} + 2^{x} + 2^{2}) d x$

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

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