Provide an appropriate response.
-Find the area lying above the x-axis and under the parabola . (Round answers to three decimal places.)
A) 32
B) 10.667
C) - 10.667
D) 11

Question

Provide an appropriate response.
-Find the area lying above the x-axis and under the parabola  . (Round answers to three decimal places.)
A) 32 
B) 10.667 
C) - 10.667 
D) 11

Quizplus · Accepted Answer

To find the area above the x-axis and under the parabola, we need to integrate the equation of the parabola from the x-intercepts. The x-intercepts are found by setting the equation equal to zero and solving for x:

0 = -2x^2 + 44x - 96

Using the quadratic formula:

x = ( -44 ± sqrt(44^2 - 4(-2)(-96)))/(-4)

x = ( -44 ± 20)/(-4)

x = 6 or 8

Therefore, we integrate the equation from x = 6 to x = 8:

∫[6,8](-2x^2 + 44x - 96)dx

= [-2/3 x^3 + 22x^2 - 96x] from x = 6 to x = 8

= (-2/3)(8^3 - 6^3) + 22(8^2 - 6^2) - 96(8 - 6)

= 10.667

Rounding to three decimal places, the answer is B) 10.667. To find the area under the curve, we need to integrate the function from its x-intercepts, which are (-3,0) and (1,0):

∫ from -3 to 1 of (x^2 + 3x + 2) dx = [x^3/3 + (3/2)x^2 + 2x] from -3 to 1

= [(1^3/3 + (3/2)(1)^2 + 2(1)) - (-3^3/3 + (3/2)(-3)^2 + 2(-3))]

= 10.667

Therefore, the answer is B.

Provide an Appropriate Response