wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Sketch the region bounded by the curves y=5x2 and y=|x1| and find its area using integration.

Open in App
Solution

The given curves are :

x2+y2=5

[ y=5x2y2=5x2x2+y2=5]

and y={1x, if x<1x1, if x>1

The required region is shown as shaded in the following figure :



y=x1 meets x2+y2=5 at B (2, 1).

y=1x meets x2+y2=5 at C (1, 2)

y=x1 and y=1x meet at A (1, 0).

Reqd. area = area (MCBLM)- area (CMAC)- area (ALBA)

= 21 5x2dx 11(1x)dx 21 (x1)dx

= [x5x22+52sin1x5]21[xx22]11[x22x]21

= [(1+52sin125)(12×2+52sin1(15))]

[(112)(112)][(22)(121)]

= 1+52sin125+152sin1(15)212

= 12+52[sin125sin1(15)]

= 12+52[sin125+sin1(15)]

sin1x+sin1y=sin1[x.1y2]+sin1[y.1x2]
x0, y 0 x2+y2 0

= 12+52[sin1(25115+15145)]

= 12+52[sin1(45+15)]

= 12+52[sin11]=12+52×π2

= (12+5π4) sq. units

flag
Suggest Corrections
thumbs-up
0
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Area under the Curve
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon