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

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

Ida Perry · Accepted Answer

Step 1  Consider a thin vertical strip of width dx.  Its length would be  =x−(x−2)  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−(x−2)]dx  The points x=0 and x=4 are the x coordinates of the points of intersection of the two curves. =∫04[x+(x−2)]dx  =∫0423×x32−x22+2x  Applying the limits  =23[432−032]−12[42−02]+2[4−0]  =23[8−0]−12[16−0]+8  =163−8+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+6−x3  for x=0 to x=2  So, Shaded Area  =∫−40[3x2+6]dx+∫02[x+6−x3]dx  =∫−40[3x24+6x]+∫02x22+6x−x44  Applying the limits  =34[02−(−4)2]+6[0−(−4)]

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