Question

Determine the area of the given region under the curve (1/x^4) [1,2]

Answer 1

Answer:

$\displaystyle \frac{7}{24}$

General Formulas and Concepts:

Algebra I

• Exponential Rule [Rewrite]: $\displaystyle b^{-m} = \frac{1}{b^m}$

Calculus

[Area] Limits of Riemann's Sums - Integrals

Integration Rule [Reverse Power Rule]:                                                                    $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

Integration Rule [Fundamental Theorem of Calculus 1]:                                          $\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)$

Step-by-step explanation:

Step 1: Define

$\displaystyle f(x) = \frac{1}{x^4} \\ \ [1, 2]$

Step 2: Find Area

1. [Integral] Set up area:                                                                                    $\displaystyle \int\limits^2_1 {\frac{1}{x^4}} \, dx$
2. [Integral] Rewrite:                                                                                            $\displaystyle \int\limits^2_1 {x^{-4}} \, dx$
3. [Integral] Reverse Power Rule:                                                                      $\displaystyle \frac{-1}{3x^3} \bigg| \limits^2_1$
4. [Area] Fundamental Theorem of Calculus:                                                   $\displaystyle \frac{7}{24}$

Topic: Calculus

Unit: Basic Integration/Riemann Sums

Book: College Calculus 10e