Find the area of the region bounded by the graph of the function f and the lines , , and .
B)
C)
D)
E)

Question

Find the area of the region bounded by the graph of the function f and the lines  ,  , and  . ​  ​
B) ​ 
C) ​ 
D) ​ 
E) ​

Quizplus · Accepted Answer

To find the area of the region bounded by the graph of the function and the given lines, we need to first find the intersection points of the function with the lines.

Setting the function equal to each line, we get:

f(x) = x^2 - 2x + 2 
x + y = 3  -->  y = 3 - x  -->  f(x) = 3 - x 
x - y = -2  -->  y = x + 2  -->  f(x) = x + 2

Setting the first two equations equal to each other, we get:

x^2 - 2x + 2 = 3 - x 
x^2 - x - 1 = 0

Using the quadratic formula, we get:

x = (1 +/- sqrt(5))/2

Setting the first and third equations equal to each other, we get:

x^2 - 2x + 2 = x + 2 
x^2 - 3x = 0

Solving for x, we get:

x = 0 or x = 3

Thus, the intersection points are:

(0, 2), ((1 + sqrt(5))/2, (5 - sqrt(5))/2), ((1 - sqrt(5))/2, (5 + sqrt(5))/2), and (3, 1)

We can see that the region we need to find the area of is a triangle with base 3 and height 1, plus a trapezoid with base (1 + sqrt(5))/2 and (1 - sqrt(5))/2, and height (5 - sqrt(5))/2 - 2 (which is the difference between the y-coordinates of the intersection points of the function with the two lines).

Using the formula for the area of a triangle and a trapezoid, we get:

Area of triangle = (1/2)*3*1 = 3/2

Area of trapezoid = [(1/2)*((1 + sqrt(5))/2 + (1 - sqrt(5))/2)]*((5 - sqrt(5))/2 - 2) = (5 - sqrt(5))/4

Total area = Area of triangle + Area of trapezoid = 3/2 + (5 - sqrt(5))/4 = (11 + sqrt(5))/4

Therefore, the answer is A) 11eb651f_cb0d_7424_b0a1_e9a6ec4d6ae4_TB8255_11.

Find the Area of the Region Bounded by the Graph