Find the area of the shaded region. ? square units 19610603461.jpg Find the

puntgewelb5

puntgewelb5

Answered question

2021-11-21

Find the area of the shaded region.
? square units

Find the area of the shaded region.
? square units

Answer & Explanation

Ida Perry

Ida Perry

Beginner2021-11-22Added 20 answers

Step 1
Consider a thin vertical strip of width dx.
Its length would be
=x(x2)
Now consider the shaded region be full of these vertical strips. So, the area of shaded region would be equal to the total area of these strips which can be calculated by using definite integrals as follows:
Area =04[x(x2)]dx
The points x=0 and x=4 are the x coordinates of the points of intersection of the two curves. =04[x+(x2)]dx
=0423×x32x22+2x
Applying the limits
=23[432032]12[4202]+2[40]
=23[80]12[160]+8
=1638+8
=163
Step 2
Similarly divide the area into thin strips as done in case one above. The only difference would be that the length of the strips would be
y=x+6(x2)=x+6+x2=3x2+6
for x=4 to x=0 and
y=x+6x3
for x=0 to x=2
So, Shaded Area
=40[3x2+6]dx+02[x+6x3]dx
=40[3x24+6x]+02x22+6xx44
Applying the limits
=34[02(4)2]+6[0(4)]

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