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Calculus: Line Integrals

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Jeffrey Jordon

Jeffrey Jordon

Expert2021-10-20Added 2605 answers

Line integrals can be used to evaluate integrals of two- or three-dimensional curves
The curve C is expressed by the following parametric equations given by

x=t2 and y=2t

Thus, to evaluate the following integral given by



we use the following line integral formula given by




Since x(t)=t2 and y(t)=2t

We want to evaluate cxyds as follows

Firstly, we must find the integrand f(x(t),y(t)) as follows




Also, the derivatives of x(t) and y(t) must be

ddtx(t)=ddt(t2)=2t and ddty(t)=ddt(2t)=2

Since 0t5, so that the upper and lower limits of the integral must be 

a=0 and b=5

Thus, using the line integral formula, we get






To evaluate the following integral 05t3(t2+1)dt, we use the method of substitution as follows. 

Let u=t2+1, so that

Differentiable both sides of the following equation u=t2+1, implies that

du=2tdt and tdt=du2

Also, we have t2=u1

Also, we find that the upper and lower limits of the integral must be

t=0u=02+1=1 and t=5u=52+1=26

Thus, the line integral becomes





Thus, the following line integral implies that





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