What is the area of the region bounded by the curve y=square root(x-1), y axis and y=1 to y=5 ?

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The area of the region bounded by the curve y = sqrt (x - 1), the y- axis and the lines y = 1 and y = 5 is the limited integral of the expression of x in terms of y, between y = 5 and y = 1.

y = sqrt (x - 1)

=> y^2 = x - 1

=> x = y^2 + 1

Int [ x dy] = Int [ y^2 + 1 dy]

=> y^3 / 3 + y + C

Between y = 1 and y = 5,

=> 5^3 / 3 + 5 + C - 1^3/3 - 1 - C

=> 124/3 + 4

=> 136/3

The required area is 136/3

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We notice that in this case, we'll have to write a function of y.

We'll put y = sqrt(x-1).

We'll raise to sqare both sides to eliminate the square root:

y^2 = x - 1

We'll add 1 both sides:

x = y^2 + 1

The area bounded by the curve is:

Int f(y)dy = F(5) - F(1) (Leibniz-Newton formula)

Int f(y)dy = Int (y^2 + 1)dy

Int (y^2 + 1)dy = y^3/3 + y + C

F(5) = (5^3/3 + 5) = 140/3

F(1) = 1/3 + 1 = 4/3

Int f(y)dy = 140/3 - 4/3 = 136/3

The area bounded by the given curve, y axis and the lines y = 1 to y = 5 is: A = 136/3 square units.

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