For a0, a curve y=f(x) passes through P(1,1). Its normal at P is a(y-1)+(x-1)=0, and at every point its tangent slope is proportional to its ordinate. Find the area bounded by the curve, the y-axis a

Question

For a>0, a curve y=f(x) passes through P(1,1). Its normal at P is a(y-1)+(x-1)=0, and at every point its tangent slope is proportional to its ordinate. Find the area bounded by the curve, the y-axis and the normal at P.

TestHub · Accepted Answer

The normal is y = 1 + ( 1 − x ) / a . Thus the area is ∫ 0 1 ​ [ 1 + ( 1 − x ) / a − e a ( x − 1 ) ] d x = a 1 ​ ( a + 2 1 ​ − 1 + e − a ) = a 1 ​ ( a − 2 1 ​ + e − a ) .

Mathematics - Area Under Curves Question with Solution | TestHub

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