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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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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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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