Answer:
\(\displaystyle 2 + \int\limits^6_2 {g(x)} \, dx = 18\)
General Formulas and Concepts:
Calculus
Integration
Integration Property [Flipping Integral]: \(\displaystyle \int\limits^b_a {f(x)} \, dx = -\int\limits^a_b {f(x)} \, dx\)
Integration Property [Splitting Integral]: \(\displaystyle \int\limits^c_a {f(x)} \, dx = \int\limits^b_a {f(x)} \, dx + \int\limits^c_b {f(x)} \, dx\)
Step-by-step explanation:
Step 1: Define
Identify
\(\displaystyle \int\limits^8_2 {g(x)} \, dx = 13\)
\(\displaystyle \int\limits^8_6 {g(x)} \, dx = -3\)
\(\displaystyle 2 + \int\limits^6_2 {g(x)} \, dx\)
Step 2: Integrate
- [Integral] Rewrite [Integration Property - Flipping Integral]: \(\displaystyle \int\limits^8_6 {g(x)} \, dx = -3 \rightarrow \int\limits^6_8 {g(x)} \, dx = 3\)
- [Integral] Rewrite [Integration Property - Splitting Integral]: \(\displaystyle 2 + \int\limits^6_2 {g(x)} \, dx = 2 + \int\limits^8_2 {g(x)} \, dx + \int\limits^6_8 {g(x)} \, dx\)
- [Integrals] Substitute: \(\displaystyle 2 + \int\limits^6_2 {g(x)} \, dx = 2 + 13 + 3\)
- Simplify: \(\displaystyle 2 + \int\limits^6_2 {g(x)} \, dx = 18\)
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
