Line Integral Calculation: F = (x^2, Xy) From (0,0) To (2,2)

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Line Integral Calculation: F = (x^2, xy) from (0,0) to (2,2)

Let's dive into calculating a line integral, guys! We've got a vector field F = (x², xy) and we need to find the line integral of F along the straight line segment from the origin (0,0) to the point (2,2). This type of problem pops up frequently in physics and engineering, helping us understand things like work done by a force along a path. It might sound intimidating, but we'll break it down step by step, making it super clear. So, buckle up, and let's get started!

Understanding Line Integrals

Before we jump into the specific problem, let's quickly recap what a line integral actually is. Simply put, a line integral calculates the integral of a function along a curve. In our case, the function is a vector field F, and the curve is a straight line. Imagine you're walking along this line, and at each point, there's a force acting on you (represented by the vector field). The line integral tells you the total effect of that force along your path. Think of it as the accumulation of the component of the force acting in the direction you're moving.

Mathematically, the line integral of a vector field F along a curve C is written as:

∫C F ⋅ dr

Where:

  • F is the vector field.
  • dr is an infinitesimal displacement vector along the curve C.
  • The dot (â‹…) represents the dot product between the vectors F and dr.

To evaluate this integral, we need to parameterize the curve C, meaning we need to express the coordinates (x, y) of points on the curve as functions of a single parameter, usually denoted by t. Then, we can rewrite the integral in terms of t and evaluate it using ordinary calculus. Don't worry if this sounds complicated; we'll walk through the whole process in our specific example.

Parameterizing the Curve

Parameterization is key. We need to describe the line segment from (0,0) to (2,2) using a parameter, say t. Since it's a straight line, we can express the x and y coordinates as linear functions of t. We want t to vary from 0 to 1, so when t = 0, we're at (0,0), and when t = 1, we're at (2,2).

So, we can write:

x = 2t y = 2t

This means our position vector r(t) is given by:

r(t) = (2t, 2t)

Now we need to find the derivative of r(t) with respect to t, which is dr/dt. This will give us the tangent vector to the curve at each point.

dr/dt = (2, 2)

So, dr = (2, 2) dt

Expressing the Vector Field in Terms of t

Next, we need to express our vector field F in terms of the parameter t. Remember, F = (x², xy). We already know that x = 2t and y = 2t, so we can substitute these into the expression for F:

F(t) = ((2t)², (2t)(2t))

F(t) = (4t², 4t²)

Calculating the Dot Product

Now we can calculate the dot product of F(t) and dr: Remember, the dot product of two vectors (a, b) and (c, d) is ac + bd.

F(t) ⋅ dr = (4t², 4t²) ⋅ (2, 2) dt

F(t) ⋅ dr = (4t² * 2 + 4t² * 2) dt

F(t) ⋅ dr = (8t² + 8t²) dt

F(t) ⋅ dr = 16t² dt

Evaluating the Integral

Alright, we're in the home stretch! Now we just need to evaluate the integral. We know that t varies from 0 to 1, so our integral is:

∫C F ⋅ dr = ∫01 16t² dt

This is a simple definite integral. Let's find the antiderivative of 16t²:

∫ 16t² dt = (16/3)t³ + C

Now we evaluate this antiderivative at the limits of integration (1 and 0):

[(16/3)(1)³] - [(16/3)(0)³] = (16/3) - 0 = 16/3

So, the line integral of F along the line segment from (0,0) to (2,2) is 16/3.

Final Answer

Therefore, the correct answer is C. 16/3

Key Takeaways

  • Parameterization is Crucial: The first and most important step is to correctly parameterize the curve. This allows you to express everything in terms of a single variable and makes the integration process much easier. Always double-check your parameterization to make sure it accurately represents the curve and its orientation.
  • Dot Product is Your Friend: The dot product is what connects the vector field to the direction of the curve. It isolates the component of the force that's acting along the path, which is what the line integral measures. Make sure you understand how to calculate the dot product correctly.
  • Don't Forget the Limits of Integration: The limits of integration are determined by the parameterization. If your parameter t ranges from 0 to 1, then those are your limits. If you change the parameterization, the limits will change accordingly.
  • Practice Makes Perfect: Line integrals can seem tricky at first, but with practice, they become much easier. Work through as many examples as you can to build your intuition and confidence.

Additional Tips for Solving Line Integral Problems

Here are some extra tips to help you ace those line integral problems, guys:

  • Sketch the Curve: Whenever possible, sketch the curve over which you're integrating. This can give you a visual understanding of the problem and help you choose the best parameterization. It's especially useful for understanding the direction of integration.
  • Consider Different Parameterizations: Sometimes, there's more than one way to parameterize a curve. Experiment with different parameterizations to see which one makes the integral easiest to evaluate. For example, you might be able to use trigonometric functions to parameterize a circle or ellipse.
  • Look for Symmetries: If the vector field or the curve has some symmetry, you might be able to simplify the calculation. For example, if the vector field is conservative, the line integral only depends on the endpoints of the curve, not the path itself.
  • Break the Curve into Pieces: If the curve is complicated, you can break it into smaller, simpler pieces and calculate the line integral over each piece separately. Then, add up the results to get the total line integral.
  • Check Your Answer: After you've calculated the line integral, take a moment to check your answer. Does it make sense in the context of the problem? Are the units correct? If possible, try to find an alternative way to calculate the line integral to verify your result.

Common Mistakes to Avoid

Let's look at some common pitfalls to dodge when you're tackling line integrals:

  • Incorrect Parameterization: This is the most common mistake. If your parameterization is wrong, everything else will be wrong too. Double-check that your parameterization accurately represents the curve and its orientation.
  • Forgetting the dt: Remember to include the dt in the integral. It's part of the differential and is essential for getting the correct answer.
  • Incorrectly Calculating the Dot Product: Make sure you know how to calculate the dot product correctly. It's easy to make a mistake, especially with more complicated vector fields.
  • Using the Wrong Limits of Integration: The limits of integration must correspond to the parameterization you're using. If you change the parameterization, you'll need to adjust the limits accordingly.
  • Not Simplifying: Before integrating, simplify the expression as much as possible. This can save you a lot of time and effort.

By avoiding these common mistakes, you'll be well on your way to mastering line integrals!

So there you have it, guys! A comprehensive guide to calculating line integrals, complete with explanations, examples, tips, and common pitfalls to avoid. Keep practicing, and you'll become a line integral pro in no time! Remember to break down the problem into smaller steps, double-check your work, and don't be afraid to ask for help if you get stuck. Good luck, and happy integrating!