MYTUTOR SUBJECT ANSWERS

270 views

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 ±√(16-4*1*3)] /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

Luis Alberto A. A Level Physics tutor, IB Physics tutor, A Level Span...

7 months ago

Answered by Luis Alberto, an A Level Maths tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

265 SUBJECT SPECIALISTS

£22 /hr

Ayyub A.

Degree: Mathematics Msci (Bachelors) - Bristol University

Subjects offered:Maths, Further Mathematics + 1 more

Maths
Further Mathematics
.MAT.

“Hi I'm Ayyub, current student at Bristol university studying Maths. I have significant tutoring experience in maths and hopefully i'll be able to help you out.”

£20 /hr

Chris B.

Degree: Economics (Bachelors) - Durham University

Subjects offered:Maths, Further Mathematics + 2 more

Maths
Further Mathematics
Economics
-Personal Statements-

“About me: I am currently in my first year of studying Economics at Durham University. I have been tutoring maths for about two years now, so have had plenty of experience passing on my knowledge and interest in my subjects to younger ...”

£22 /hr

Max G.

Degree: Mathematics and Physics (Masters) - Durham University

Subjects offered:Maths, Physics+ 2 more

Maths
Physics
Chemistry
Biology

“I am a maths and physics student at the University of Durham, for as long as I can remember i have been obsessed with all things science! I am patient, friendly and most of all understanding to the fact that the sciences aren't for ev...”

About the author

£20 /hr

Luis Alberto A.

Degree: Mathematical Modelling (Masters) - University College London University

Subjects offered:Maths, Spanish+ 1 more

Maths
Spanish
Physics

“I am keen to express my interest in providing academic support. I am recently graduated with distinction from the Mathematical Modelling MSc programme at University College London (UCL). As an undergraduate, I studied a BSc in Physics...”

MyTutor guarantee

You may also like...

Other A Level Maths questions

Find the area under the curve y = (4x^3) + (9x^2) - 2x + 7 between x=0 and x=2

Differentiate: y=12x(2x+1)+1/x

Differentiate xcos(x) with respect to x

How to integrate by parts

View A Level Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok