# Calculate the work done by a particle under the influence of a force #y^2 hat(i) - x^2hat(j)# along the curve #y=4x^2# from #(0,0)# to #(1,4)#?

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# int_C vec(F) * d vec(r) = 1.2#

The integral is known as a line integral.

So we have:

And:

Hence,

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To calculate the work done by the particle along the curve ( y = 4x^2 ) from (0,0) to (1,4), we first need to parameterize the curve. Let's denote ( x = t ) such that ( y = 4t^2 ), where ( t ) ranges from 0 to 1. Then, differentiate ( y ) with respect to ( t ) to find ( dx ) and ( dy ) in terms of ( dt ).

Next, express the given force as ( F = y^2 \mathbf{i} - x^2 \mathbf{j} ), and substitute the parameterized values of ( x ) and ( y ).

After that, integrate ( F \cdot dr ) along the curve from the initial point (0,0) to the final point (1,4), where ( dr ) is the differential displacement vector ( dx \mathbf{i} + dy \mathbf{j} ).

Finally, evaluate the integral to find the work done by the force along the given curve.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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