A line has equation y = 2x + c and a curve has equation y = 8 − 2x − x^2, if c=11 find area between the curves

Extracted from Cambridge International Examinations / MATHEMATICS 9709/11 Paper 1 Pure Mathematics 1

(7 points out of 10 for this question)

A line has equation y = 2x + c and a curve has equation y = 8 − 2x − x². For the case where c = 11, find the x-coordinates of the points of intersection of the line and the curve. Find also, by integration, the area of the region between the line and the curve.

SOLUTION

y=2x + c    …...       (1)

y=8-2x-x² …..         (2)

c=11, then y=2x+11

and the intersection is given when (1) = (2)

that is 2x + 11 = 8-2x-x²

then we obtain a 2nd order polynomial by putting all to the LHS

x² + 4x +3 =0 

which is equivalent to write 

(x+3)(x+1)=0 

This means that the intersection the solution of this root. Since we have factors,  it is fast to check that x=-3,-1. 

But just to check we solve this polynomial using the Bernoulli’s approach:

 x_{1,2}= [-4 ±√(16-413)] /2

            =( -4 ± 2)/2 = -3, -1

Then the x-coordinates of the points of intersection of the line and the curve are : x_{1,2}= -3, -1

Now, the area of the region between the line and the curve is the same as "the area of the curve minus the area of the line”. 

Thus, by integration of  (2)-(1) using the limits of integration  x_{1,2}=-3,-1

∫ (8-2x-x²)dx - ∫ (2x+11)dx =  area of the region between the line and the curve 

 =[ 8x -x²- x³/3] - [x²+11]

Apply the limits  x_{1,2}=-3,-1

then, area of the region between the line and the curve =1/3