Area of Shaded Region

Edexcel IAL WMA12/01/P2/Oct 2019/Q8 (Integration)

Solutions relying on calculator technology are not acceptable in this question.

Figure 2 shows a sketch of part of a curve with equation​​ 

y=8x-52x3            x>0

The region R, shown shaded in Figure 2, is bounded by the curve, the line with equation x = 2, the x-axis and the line with equation x = 4​​ 

Find the exact area of R.​​ 

(5)

• Find the value of the constant k such that​​ 

∫-3612x2+kdx=55

(4)

SOLUTION​​ 

i- To find the area of R, lets integrate the equation of the curve with the upper limit as 4 and lower limit as 2. ​​ 

So lets first split the expression.

∫248x-52x2dx=∫548x122x2-52x2dx

=∫244x-32-52x-2dx

=4x-12-12-5x-1-224

=-8x-12+52x24

=-8x+52x24

Applying limits

=-84+524--82+522

=-82+58+82-54

=-328+58-108+82 x 22 

=-378+822

∫248x-52x2dx=42-378

ii- To find the value of k, lets simply integrate.​​ 

∫-3612x2+kdx=55

x36+kx-36=55

636+6k--336-3k

36+6k--276-3k=55

36+6k+276+2k=55

9k+812=55

9k=55-812

9k=1102-812

9k=292

k=2918