What is the area of the region between the graphs of f(x)=x^(2)+2x and g(x)=2x+1 ? Choose 1 answer: (A) (2)/(3) (B) 2 (C) (4)/(3) (D) (10)/(3)

Set Equations Equal: To find the area between the two curves, we first need to find the points of intersection by setting f(x) equal to g(x).Solve for x: Set f(x) = g(x): x^2 + 2x = 2x + 1.Calculate Limits of Integration: Subtract 2x from both sides to simplify the equation: x^2 = 1.Set up Integral: Take the square root of both sides to find the values of x: x = ±1.Substitute Functions: The points of intersection are x = -1 and x = 1. These will be the limits of integration to find the area between the curves.Simplify Integrand: Set up the integral to calculate the area: A = ∫ from -1 to 1 of (g(x) - f(x)) dx.Integrate Function: Substitute the functions into the integral: A = ∫ from -1 to 1 of ((2x + 1) - (x^2 + 2x)) dx.Evaluate Upper Limit: Simplify the integrand: A = ∫ from -1 to 1 of (1 - x^2) dx.Evaluate Lower Limit: Integrate the function: A = [x - (1/3)x^3] from -1 to 1.Find Total Area: Evaluate the integral at the upper limit: A(1) = 1 - (1/3)(1)^3 = 1 - 1/3 = 2/3.Final Answer: Evaluate the integral at the lower limit: A(-1) = -1 - (1/3)(-1)^3 = -1 + 1/3 = -2/3.Find the difference between the two evaluations to get the total area: A = A(1) - A(-1) = 2/3 - (-2/3) = 2/3 + 2/3 = 4/3.The area of the region between the graphs of f(x) and g(x) is 4/3, which corresponds to answer choice (C).

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What is the area of the region between the graphs of
f(x)=x^(2)+2x and
g(x)=2x+1 ?
Choose 1 answer:
(A)
(2)/(3)
(B) 2
(C)
(4)/(3)
(D)
(10)/(3)

What is the area of the region between the graphs of $f(x)=x^{2}+2 x$ and $g(x)=2 x+1$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{2}{3}$ $\newline$ (B) $2$ $\newline$ (C) $\frac{4}{3}$ $\newline$ (D) $\frac{10}{3}$

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Math Problems

Calculus

Intermediate Value Theorem

Full solution

Q. What is the area of the region between the graphs of

f(x)=x^{2}+2 x

and

g(x)=2 x+1

\newline

Choose

1

answer:

\newline

(A)

\frac{2}{3}

\newline

(B)

2

\newline

(C)

\frac{4}{3}

\newline

(D)

\frac{10}{3}

Set Equations Equal: To find the area between the two curves, we first need to find the points of intersection by setting $f(x)$ equal to $g(x)$ .
Solve for x: Set $f(x) = g(x)$ : $x^2 + 2x = 2x + 1$ .
Calculate Limits of Integration: Subtract $2x$ from both sides to simplify the equation: $x^2 = 1$ .
Set up Integral: Take the square root of both sides to find the values of $x$ : $x = \pm 1$ .
Substitute Functions: The points of intersection are $x = -1$ and $x = 1$ . These will be the limits of integration to find the area between the curves.
Simplify Integrand: Set up the integral to calculate the area: $A = \int_{-1}^{1} (g(x) - f(x)) \, dx$ .
Integrate Function: Substitute the functions into the integral: $A = \int_{-1}^{1} ((2x + 1) - (x^2 + 2x)) \, dx$ .
Evaluate Upper Limit: Simplify the integrand: $A = \int_{-1}^{1} (1 - x^2) \, dx$ .
Evaluate Lower Limit: Integrate the function: $A = [x - (\frac{1}{3})x^3]$ from $-1$ to $1$ .
Find Total Area: Evaluate the integral at the upper limit: $A(1) = 1 - (\frac{1}{3})(1)^3 = 1 - \frac{1}{3} = \frac{2}{3}$ .
Final Answer: Evaluate the integral at the lower limit: $A(-1) = -1 - \frac{1}{3}(-1)^3 = -1 + \frac{1}{3} = -\frac{2}{3}$ .
Final Answer: Evaluate the integral at the lower limit: $A(-1) = -1 - (\frac{1}{3})(-1)^3 = -1 + \frac{1}{3} = -\frac{2}{3}$ .Find the difference between the two evaluations to get the total area: $A = A(1) - A(-1) = \frac{2}{3} - (-\frac{2}{3}) = \frac{2}{3} + \frac{2}{3} = \frac{4}{3}$ .
Final Answer: Evaluate the integral at the lower limit: $A(-1) = -1 - (1/3)(-1)^3 = -1 + 1/3 = -2/3$ .Find the difference between the two evaluations to get the total area: $A = A(1) - A(-1) = 2/3 - (-2/3) = 2/3 + 2/3 = 4/3$ .The area of the region between the graphs of $f(x)$ and $g(x)$ is $4/3$ , which corresponds to answer choice (C).

More problems from Intermediate Value Theorem

Question

Let

f

be a continuous function on the closed interval

[1,5]

, where

f(1)=1

What is the area of the region between the graphs of f(x)=x2+2x f(x)=x^{2}+2 x f(x)=x2+2x and g(x)=2x+1 g(x)=2 x+1 g(x)=2x+1 ?\newlineChoose 111 answer:\newline(A) 23 \frac{2}{3} 32​\newline(B) 222\newline(C) 43 \frac{4}{3} 34​\newline(D) 103 \frac{10}{3} 310​

What is the area of the region between the graphs of $f(x)=x^{2}+2 x$ and $g(x)=2 x+1$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{2}{3}$ $\newline$ (B) $2$ $\newline$ (C) $\frac{4}{3}$ $\newline$ (D) $\frac{10}{3}$