Area of Shaded Region

Edexcel IAL WMA12/01/P2/Jan 2020/Q4 (Integration)

Figure 1 shows a sketch of the curve with equation​​ 

y=2x3+7             x≥0

The finite region R, shown shaded in Figure 1, is bounded by the curve, the y - axis and the line with equation y = 17​​ 

Find the exact area of R.​​ 

(6)

SOLUTION​​ 

To find the area of region R, we need to subtract the area under the curve from the area of rectangle as shown below in the figure. But, for all this we need to know the width of rectangle and the upper limit of the area enclosed, which are actually same. Therefore, we have to first find the x cordinate of the point of intersection of curve and line and​​ then substitute it as the upper limit, while finding the area under the curve, and the width, while finding the area of the ractangle.​​ 

First, equating both equations of curve and line to find the x-cordinate of point of intersection.​​ 

ycurve=yline

2x2+7=17

2x2=10

x2=5

x=±5

We will choose the positive value since point of intersection is in first quadrant.​​ 

x=5

Now,​​ 

Area of R=Area of rectangle-Area under the curve

Area of R=lb-∫05ycurve.dx

Area of R=17(5)-∫05(2x2+7).dx

Area of R=175-2x33+7x05

Now, applying​​ ​​ limits.

Area of R=175-2533+7(5)-0

Area of R=175-3153

Area of R=2053