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Question

Using integration, find the area of the region bounded by the curves y=5x2 and y=|x1|


Solution


We have y=5x2(i) and y=|x1|={x1,ifx11x,ifx<1(ii)

Solving (i) and (ii), |x1|2=[5x2]2
x22x+1=5x2x2x2=0y=|x1|(x+1)(x2)=0x=1,2[1]

Required area =21yidx21yiidx        [2]

=215x2dx21|x1|dx     [2]

=215x2dx[11|x1|dx+21|x1|dx]

=21=5x2dx11(x1)dx21(x1)dx

=[x25x2+52sin1x5]21+[(x1)22]11[(x1)22]21

=[x25x2+52sin1x5]21[(x1)22]11[(x1)22]21

[1+52sin125][152sin115]+0212+0

52[sin125+sin115]12 sq. units

52[sin1(25115+1545)]12

52[sin1(45+15)]12=52×π212=(5π412) sq. units.
 

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