What is the area of the region between the graphs of f(x) = x^2 + 2xf(x) = x2 +2x, and g(x) = 2x+1g(x) = 2x+1g(x) = 2x + 1?

enclosed area A = 2/3 units^2

Step-by-step explanation:

Given:-

- The two functions are given:-

                  f ( x ) = x^2 + 2x

                  g ( x ) = 2x + 1

Find:-

What is the area of the region between the graphs of f(x) and g(x)

Solution:-

- We will determine the interval or limits in which the two given function f(x) and g(x).

- We will find the interval by determining the point of intersection between the two graphs. Hence,

                  f ( x ) = g ( x )

                  x^2 + 2x = 2x + 1

                  x^2 = 1

                  x = ± 1 , y = 3 , -1

- The pair of intersection coordinates are ( -1 , -1 ) and ( 1 , 3 ). The functions are bounded by the intervals:

                  -1 ≤ x ≤ 1

                  -1 ≤ y ≤ 3

- Now sketch each function to visualize the enclosed region. ( See attachment ). We see that in the range ( -1 ≤ x ≤ 1 ) the function g ( x ) > f ( x ).

- So we will find the area along the y-direction with the interval ( -1 ≤ x ≤ 1 ) will define the limits of integration. The enclosed area is given by:

- The limits of integration are  [ a , b ] = [ 1 , -1 ].

- The enclosed area A = 2/3 units^2.

infinitely many solutions

step-by-step explanation:

we have been given the equation;

2(2x + 3) = -4 + 2(2x + 5)

we need to determine the value of x.

2(2x + 3), we expand this expression to obtain 4x + 6

similarly 2(2x + 5) will become 4x + 10

now we have;

4x + 6 = -4 + 4x + 10

4x + 6 = 4x + 10 - 4

4x + 6 = 4x + 6

the expression on the left hand side is identical to the one on the right hand side. therefore, we are looking at just a single straight line whose equation is;

y = 4x + 6

consequently, we shall have infinitely many solutions

the correct answer to your question is a. 4^4

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