Use the Midpoint Rule with the given value of n to to approximate the integral. Round the answer...

Question:

Use the Midpoint Rule with the given value of n to to approximate the integral. Round the answer to four decimal places.

{eq}\int_{2}^{16}\sqrt{x^{4}+3}\ dx,\ n=4 {/eq}

Integral Approximation:

To solve this problem, we'll use the following theorems:

1.) Midpoint Rule:

{eq}\int_a^{b} f(x) \, dx \approx \, M_n = \Delta x \left[ f(\bar x_1) + f(\bar x_2) + ......+ f(\bar x_n) \right] {/eq}

Where,

{eq}\Delta x = \frac{b- a}{n} {/eq}

And

{eq}\bar x_i = \frac{1}{2} (x_{(i-1)} + x_i) = \, \, midpoint \, of \, \, \left[ x_{(i-1)},x_i \right] {/eq}

Answer and Explanation:

We are given:

{eq}n = 4 \, , \, \Delta x = \frac{16- 2}{4} = 3.5 \, ,\, a = 2 \, and \, b = 16 {/eq}

Let us assume that {eq}f(x) =...

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Approximating Definite Integrals on a Graphing Calculator
Approximating Definite Integrals on a Graphing Calculator

from Saxon Calculus Homeschool: Online Textbook Help

Chapter 7 / Lesson 8
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