Arc of a Curve

Calculus Level pending

Let C C be the arc of the curve y = x y=\sqrt{x} from points ( 1 , 1 ) (1, 1) to ( 4 , 2 ) (4, 2) . What is the value of N = ∫ C ( x 2 y 3 − x ) d y ? N = \int_C (x^2y^3-\sqrt{x})\ dy?

30.375 15.188 60.75 121.5

Mahindra Jain by Mahindra Jain

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1 solution

Tom Engelsman
Nov 8, 2020

This path integral is equivalent to:

N = ∫ 1 2 ( x ) 4 y 3 − x d y = ∫ 1 2 y 7 − y d y = y 8 8 − y 2 2 ∣ 1 2 = ( 32 − 2 ) − ( 1 8 − 1 2 ) = 30.375 . N = \int_{1}^{2} (\sqrt{x})^4 y^3 - \sqrt{x} dy = \int_{1}^{2} y^7 - y dy = \frac{y^8}{8} - \frac{y^2}{2}|_{1}^{2} = (32-2)-(\frac{1}{8}-\frac{1}{2}) = \boxed{30.375}.

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