A curve has equation y = 10 + 8x + x^2 - x^3. X >= 0

a)Find the coordinates of the turning point. and show whether it is maximum or minimum.

b) Hence, Find the area of the region bounded by the curve, the line y = 11x and the y axis?

I manage to complete a) which I got :

dy/dx = 8 + 2x - 3x^2

dy/dx = 0

(-3x^2 + 2x + 8) x-1 = 0
= 3x^2 - 2x -8 = 0
(3x + 4) ( x - 2) = 0
x = -4/3 or x = 2

since x >= 0 I chose 2
x = 2
y = 10 + 8(2) + (2)^2 - (2)^3
y = 22

so turning point is (2, 22)

//finding max or min :

f(-4/3) = 10 + 8(-4/3) + (-4/3)^2 - (-4/3)^3
= -50/27, f(-4/3) < 0, max

f(2) = 10 + 8(2) + (2)^2 + (2)^3
=22, f(2) > 0, min

The part I am stuck is part b) I don't quite sure how to solve it.Please can you help me with this part? Thank you !

Find where the curves intersect. Clearly, at x=2, since y(2) = 22 = 11*2

I was wondering about the "hence".

So, the area in question has two parts. That under the line and that under the curve:

a = ∫[0,2] 11x dx + ∫[2,3.7988] 10+8x+x^2-x^3 dx
= 22 + 27.2556
= 49.2556

I'm surprised they did not come up with an integer zero for y.

To find the area of the region bounded by the curve y = 10 + 8x + x^2 - x^3, the line y = 11x, and the y-axis, you need to determine the points where the curve intersects the line y = 11x first and then integrate to find the area.

1. Find the points where the curve intersects the line y = 11x:
Set y = 11x in the equation of the curve and solve for x:

10 + 8x + x^2 - x^3 = 11x

Rearrange the equation to get it in the form of a polynomial:

x^3 - x^2 + 3x - 10 = 0

Solve this equation to find the points where the curve intersects the line. You can use numerical methods like Newton's method or use software/graphing calculators to find the solutions:

x ≈ -3.654, x ≈ 0.827, x ≈ 2.828

2. Now that you have the x-values of the points where the curve intersects the line, you can integrate the function between these points to find the area.
To find the area between two curves, you need to subtract the area under the lower curve from the area under the upper curve.

So, to find the area bounded by the curve, line, and y-axis, you need to perform the following steps:

a) Integrate the curve function with respect to x between the x-values of intersection points.
b) Integrate the line function with respect to x between the x-values of intersection points.
c) Subtract the second integral from the first integral to find the area.

Let's go through these steps:

a) Integrate the curve function:
∫(10 + 8x + x^2 - x^3) dx

Integrating term by term, we get:
= 10x + 4x^2/2 + x^3/3 - x^4/4 + C

b) Integrate the line function:
∫(11x) dx

= 11x^2/2 + C

c) Evaluate the definite integral by substituting the limits (intersection points) and subtracting:
Area = [(10x + 4x^2/2 + x^3/3 - x^4/4) - (11x^2/2)] from x = -3.654 to x = 0.827
+ [(10x + 4x^2/2 + x^3/3 - x^4/4) - (11x^2/2)] from x = 0.827 to x = 2.828

Evaluate the definite integrals using the limits and subtract to find the area.

This will give you the area of the region bounded by the curve, the line y = 11x, and the y-axis.