Mathematics, 22.03.2021 18:50 nayelieangueira

Find the area of the region enclosed by the graphs of the functions f(x)=x, g(x)=x^3
by partitioning the x- axis.
(Use symbolic notation and fractions where needed.)

Answers: 1

Show answers

Answers

Answer from: bellbug08

$\displaystyle A = \frac{1}{2}$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to RightEquality Properties

Algebra I

Terms/CoefficientsFunctionsFunction NotationGraphing

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.

Also remember that finding area and evaluating are two different things.

Step 1: Define

f(x) = x

g(x) = x³

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]

1st Integral: [-1, 0]

2nd Integral: [0, 1]

Step 3: Find Area of Region

Integration.

Substitute in variables [Area of a Region Formula]: $\displaystyle A = \int\limits^0_{-1} {[x^3 - x]} \, dx + \int\limits^1_0 {[x - x^3]} \, dx$ [Area] Rewrite Integrals [Integration Property - Subtraction]: $\displaystyle A = (\int\limits^0_{-1} {x^3} \, dx - \int\limits^0_{-1} {x} \, dx) + (\int\limits^1_0 {x} \, dx - \int\limits^1_0 {x^3} \, dx)$ [Area] [Integrals] Integrate [Integration Rule - Reverse Power Rule]: $\displaystyle A = [\frac{x^4}{4} \bigg|\limits^0_{-1} - (\frac{x^2}{2}) \bigg|\limits^0_{-1}]+ [\frac{x^2}{2} \bigg|\limits^1_0 - (\frac{x^4}{4}) \bigg|\limits^1_0]$ [Area] Evaluate [Integration Rule - FTC 1]: $\displaystyle A = [\frac{-1}{4} - (\frac{-1}{2})] + [\frac{1}{2} - \frac{1}{4}]$ [Area] [Brackets] Add/Subtract: $\displaystyle A = \frac{1}{4} + \frac{1}{4}$ [Area] Add: $\displaystyle A = \frac{1}{2}$

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

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

Book: College Calculus 10e

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

f(x)=x, g(x)=x^3
by partitionin

Answer from: Quest

225x²+4=0

225x²=0-4

225x²=-4

x²=-4/225

x=±√-4/225

Answer from: Quest

find the magnitude of the trig term

csc

(

)

−

by taking the absolute value of the coefficient.

find the lower bound of the range.

tap for more

−

find the upper bound of the range.

tap for more

the range is

≤

−

≥

(

−

∞

−

]

∪

[

∞

)

{

≤

−

≥

}

step-by-step explanation:

Another question on Mathematics

Mathematics, 20.06.2019 18:04

Sasha invests $1000 that earns 8% interest compounded annually. write a function that describes the value, v, of the investment after t years.

Answers: 1

Answer

Mathematics, 21.06.2019 18:10

Which of these sequences is a geometric sequence a 1,2,4, 7, 11, 16, 22. b. 2.4.8.14.22.38 c.3.6.9. 15.18,21 d. 3,9,27,81,243,729,

Answers: 1

Answer

Mathematics, 21.06.2019 19:30

The position of a moving particle is given by the position function: f(t)=-9t-t^2-0.2t^3+0.1t^4 0 a. at what time does the particle reverse direction? b. when is the displacement positive? (round one decimal place and answer in interval notation) c. when is the displacement negative? (round one decimal place and answer in interval notation) d. when is the particle’s acceleration positive? (round one decimal place and answer in interval notation) e. when is the particle’s acceleration negative? (round one decimal place and answer in interval notation)

Answers: 3

Answer

Mathematics, 22.06.2019 01:00

Given the numbers c = –3 and d = 4, which statement is true? a. |–c| = 3 and |–d| = –4 b. |–c| = 3 and –|d| = –4 c. –|c| = –4 and –|d| = 3 d. |–c| = 3 and –|d| = 4

Answers: 1

Answer

You know the right answer?

Find the area of the region enclosed by the graphs of the functions f(x)=x, g(x)=x^3
by parti...

Questions