Which expression is equivalent to…

Find the area of the region enclosed by the graphs of the functions

Vaselesa [24]

Answer:

$\displaystyle A = \frac{8}{21}$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets
Parenthesis
Exponents
Multiplication
Division
Addition
Subtraction

Left to Right

Equality Properties

Multiplication Property of Equality
Division Property of Equality
Addition Property of Equality
Subtraction Property of Equality

Algebra I

Terms/Coefficients
Functions
Function Notation
Graphing
Solving systems of equations

Calculus

Area - Integrals

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 [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:

*Note:

Remember that for the Area of a Region, it is top function minus bottom function.

Step 1: Define

f(x) = x²

g(x) = x⁶

Bounded (Partitioned) by x-axis

Step 2: Identify Bounds of Integration

Find where the functions intersect (x-values) to determine the bounds of integration.

Simply graph the functions to see where the functions intersect (See Graph Attachment).

Interval: [-1, 1]

Lower bound: -1

Upper Bound: 1

Step 3: Find Area of Region

Integration

Substitute in variables [Area of a Region Formula]: $\displaystyle A = \int\limits^1_{-1} {[x^2 - x^6]} \, dx$
[Area] Rewrite [Integration Property - Subtraction]: $\displaystyle A = \int\limits^1_{-1} {x^2} \, dx - \int\limits^1_{-1} {x^6} \, dx$
[Area] Integrate [Integration Rule - Reverse Power Rule]: $\displaystyle A = \frac{x^3}{3} \bigg| \limit^1_{-1} - \frac{x^7}{7} \bigg| \limit^1_{-1}$
[Area] Evaluate [Integration Rule - FTC 1]: $\displaystyle A = \frac{2}{3} - \frac{2}{7}$
[Area] Subtract: $\displaystyle A = \frac{8}{21}$

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

Unit: Area Under the Curve - Area of a Region (Integration)

Book: College Calculus 10e

6 0

3 years ago

Using the diagonals, determine whether or not ABCD is a rhombus? How do I show my work on this?

Ann [662]

Answer:

If the third term of an arithmetic sequence is-21and the sixth term is-6 find the first and eighth term of the sequence

Step-by-step explanation:

8 0

3 years ago

The cost of a banquet can be calculated using the function

kotykmax [81]

Answer:

3 people attended the banquet

Step-by-step explanation:

The function is given as:

b = 75 + 15n where

b represents the total = 122

n represents the number of people attending = ??

If the total cost of the banquet is 122, how many people attended the banquet?

Hence,

122 = 75 + 15n

122 - 75 = 15n

47 = 15n

Divide both sides by 15

n = 47/15

n = 3.1333333333

Approximately to the nearest whole number = 3

Therefore, 3 people attended the banquet.

3 0

3 years ago

Geometry multiple choice question HELLPPP

alina1380 [7]

Answer:

27/36

Step-by-step explanation:

tan C = opp/adj

For angle C, the opposite leg is 27,a nd the adjacent leg is 36.

tan C = 27/36

Answer: 27/36

4 0

3 years ago

the melting point of mercury is -39°C. the freezing point of alcohol is 114°C. How much cooker is the melting point of mercury t

8_murik_8 [283]

This is one of those cases where you have to believe that a minus times a minus is a plus.

114 - - 39 = 153 degrees. The freezing of mercury is 153 degrees cooler than the freezing point of alcohol. I would check your descriptions. Did you mean boiling point of Alcohol?

8 0

4 years ago