Question

Use the Fundamental Theorem of Calculus to find the "area under curve" of

Answer 1

Answer:

$\displaystyle A = 300$

General Formulas and Concepts:

Calculus

Integrals

• Definite Integrals
• Area under the curve
• Integration Constant C

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]:                                                             $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

Integration Property [Addition/Subtraction]:                                                           $\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx$

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

Step-by-step explanation:

Step 1: Define

Identify

f(x) = 6x + 19

Interval [12, 15]

Step 2: Find Area

1. Substitute in variables [Area of a Region Formula]:                                       $\displaystyle A = \int\limits^{15}_{12} {(6x + 19)} \, dx$
2. [Integral] Rewrite [Integration Property - Addition/Subtraction]:                   $\displaystyle A = \int\limits^{15}_{12} {6x} \, dx + \int\limits^{15}_{12} {19} \, dx$
3. [Integrals] Rewrite [Integration Property - Multiplied Constant]:                   $\displaystyle A = 6\int\limits^{15}_{12} {x} \, dx + 19\int\limits^{15}_{12} {} \, dx$
4. [Integrals] Integrate [Integration Rule - Reverse Power Rule]:                      $\displaystyle A = 6(\frac{x^2}{2}) \bigg| \limits^{15}_{12} + 19(x) \bigg| \limits^{15}_{12}$
5. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:              $\displaystyle A = 6(\frac{81}{2}) + 19(3)$
6. Simplify:                                                                                                             $\displaystyle A = 300$

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

Unit: Integration

Book: College Calculus 10e