Question

Given the hyperbola y = 1/x, find the area under the curve between x = 5 and x = 33 to 3 sig. dig. namely: integral subscript 5 superscript 33 1 over x d x space equals . State the definite integral and evaluate it:Given the hyperbola y = 1/x, find the area under the curve between x = 5 and x = 33 to 3 sig. dig. namely: integral subscript 5 superscript 33 1 over x d x space equals . State the definite integral and evaluate it:

Answer 1

`\displaystyle A = \int\limits^(33)_(5) {(1)/(x)} \, dx`

`\displaystyle A = \ln (33)/(5)`

General Formulas and Concepts:

Algebra II

• Logarithmic Properties

Calculus

Integration

• Integrals

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

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

Explanation:

Step 1: Define

Identify

`\displaystyle y = (1)/(x)`

Bounds: [5, 33]

Step 2: Find Area

1. Substitute in variables [Area of a Region Formula]:
   `\displaystyle A = \int\limits^(33)_(5) {(1)/(x)} \, dx`
2. [Integral] Integrate [Logarithmic Integration]:
   `\displaystyle A = \ln |x| \bigg| \limits^(33)_5`
3. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   `\displaystyle A = \ln |33| - \ln |5|`
4. Condense:
   `\displaystyle A = \ln (33)/(5)`

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

Unit: Integration