Question

Find the area of the region under the graph of the function f on the interval [4, 8].
f(x) = 10/x^2

If you could show the steps and the answer, I would really appreciate it!

Answer 1

Answer:

$\displaystyle \int\limits^8_4 {\frac{10}{x^2}} \, dx = \frac{5}{4}$

General Formulas and Concepts:

Calculus

Integration

• 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]:                                     $\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)$

Integration Property [Multiplied Constant]:                                                         $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

Area of a Region Formula:                                                                                     $\displaystyle A = \int\limits^b_a {[f(x) - g(x)]} \, dx$

Step-by-step explanation:

Step 1: Define

Identify

$\displaystyle f(x) = \frac{10}{x^2} \\\left[ 4 ,\ 8 \right]$

Step 2: Find Area

1. Substitute in variables [Area of a Region Formula]:                                   $\displaystyle \int\limits^8_4 {\frac{10}{x^2}} \, dx$
2. [Integral] Rewrite [Integration Property - Multiplied Constant]:                 $\displaystyle \int\limits^8_4 {\frac{10}{x^2}} \, dx = 10 \int\limits^8_4 {\frac{1}{x^2}} \, dx$
3. [Integral] Integrate [Integration Rule - Reverse Power Rule]:                     $\displaystyle \int\limits^8_4 {\frac{10}{x^2}} \, dx = 10 \bigg( \frac{-1}{x} \bigg) \bigg| \limits^8_4$
4. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:           $\displaystyle \int\limits^8_4 {\frac{10}{x^2}} \, dx = 10 \bigg( \frac{1}{8} \bigg)$
5. Simplify:                                                                                                        $\displaystyle \int\limits^8_4 {\frac{10}{x^2}} \, dx = \frac{5}{4}$

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration