Question

CALC HELP!
16
18
8
32

Answer 1

Answer:

$\displaystyle 2 + \int\limits^6_2 {g(x)} \, dx = 18$

General Formulas and Concepts:

Calculus

Integration

• Integrals

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

1. [Integral] Rewrite [Integration Property - Flipping Integral]:                       $\displaystyle \int\limits^8_6 {g(x)} \, dx = -3 \rightarrow \int\limits^6_8 {g(x)} \, dx = 3$
2. [Integral] Rewrite [Integration Property - Splitting Integral]:                       $\displaystyle 2 + \int\limits^6_2 {g(x)} \, dx = 2 + \int\limits^8_2 {g(x)} \, dx + \int\limits^6_8 {g(x)} \, dx$
3. [Integrals] Substitute:                                                                                    $\displaystyle 2 + \int\limits^6_2 {g(x)} \, dx = 2 + 13 + 3$
4. Simplify:                                                                                                         $\displaystyle 2 + \int\limits^6_2 {g(x)} \, dx = 18$

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

Unit: Integration