# Find the value of p and q and the area of the region R in the following case:A curve C is defined by y=x^3-9x^2+px, where p is a constant. The line l has equation y+2x=q, where q is a constant. The...

Find the value of p and q and the area of the region R in the following case:

A curve C is defined by y=x^3-9x^2+px, where p is a constant. The line l has equation y+2x=q, where q is a constant. The point A is the intersection of C and l, and C has a minimum at the point B. The x-coordinates of A and B are 1 and 4 respectively.

Show that p=24 and calculate the value of q

Find the area of the region R is bounded by C, l and the x-axis.

The equation of the curve C is y = x^3 - 9x^2 + px. The line y + 2x = q intersects the curve at the point where x = 1 and the curve y = x^3 - 9x^2 + px has the minimum point at x = 4.

y = x^3 - 9x^2 + px

y' = 3x^2 - 18x + p

At the minimum point y' = 0

=> 3x^2 - 18x + p = 0

As the value of x here is 4

=> 3*16 - 18*4 + p = 0

=> p = 24

As the curve and the line meet at the point where x = 1

1 - 9 + 24 = q - 2

=> q = 18

From the graphs of the curve and the line it is seen that the required area can be divided into two parts. One is the integral of the integral of the curve from 0 to 1 and the other is a right triangle. The total area is the sum of the two:

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=> ``

=> ``

=> 73.5

**The required area R is 73.5**

Sorry about an error made in the calculation of the area R.

From the graphs of the curve and the line it is seen that the required area can be divided into two parts. One is the integral of the integral of the curve from 0 to 1 and the other is a right triangle. The total area is the sum of the two:

`int_(0)^1 x^3 - 9x^2 + 24x dx + (1/2)*8*16`

=> `(1^4)/4 - 9*1^4/3 + 24*1^2/2 + 8*8`

=> `9.25 + 64`

=> 73.5

**The required area R is 73.5**