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

`\displaystyle \int\limits^8_4 {(10)/(x^2)} \, dx = (5)/(4)`

General Formulas and Concepts:

Calculus

Integration

• Integrals

Integration Rule [Reverse Power Rule]:
`\displaystyle \int {x^n} \, dx = (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`

Explanation:

Step 1: Define

Identify

`\displaystyle f(x) = (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 {(10)/(x^2)} \, dx`
2. [Integral] Rewrite [Integration Property - Multiplied Constant]:
   `\displaystyle \int\limits^8_4 {(10)/(x^2)} \, dx = 10 \int\limits^8_4 {(1)/(x^2)} \, dx`
3. [Integral] Integrate [Integration Rule - Reverse Power Rule]:
   `\displaystyle \int\limits^8_4 {(10)/(x^2)} \, dx = 10 \bigg( (-1)/(x) \bigg) \bigg| \limits^8_4`
4. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   `\displaystyle \int\limits^8_4 {(10)/(x^2)} \, dx = 10 \bigg( (1)/(8) \bigg)`
5. Simplify:
   `\displaystyle \int\limits^8_4 {(10)/(x^2)} \, dx = (5)/(4)`

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

Unit: Integration