Line Integral xy*ds along the Path C: r(t) = 4t*i + 3t*j for t in [0, 1]

TL;DR

Calculating the line integral along a given path by plugging the path equation into the line integral formula.

Transcript

hey what's up YouTube and this problem we're going to evaluate the line integral along the path C given by this function R so solution so first we call the formulas if you have a line integral of a function f of X Y DS this is equal to the definite integral from A to B of f of X of T Y of T square root and then you have X prime squared plus y prime... Read More

Key Insights

• 🫥 Line integrals can be used to calculate the total change of a function along a given path.
• 🫥 The line integral formula includes the path equation and the derivative of the path functions.
• 🫥 Integrating the line integral equation requires simplifying and applying the power rule.
• 🫥 Evaluating the line integral involves plugging in the range of t values and calculating the resulting integral.

Questions & Answers

Q: What is the formula for calculating a line integral along a given path?

The formula for a line integral of a function f(x, y) along a path C is ∫f(x, y)√(x'^2 + y'^2) dt, where x' and y' are the derivatives of x(t) and y(t) with respect to t.

Q: How do you evaluate a specific line integral using the formula?

To evaluate a specific line integral, plug the path functions x(t) and y(t) into the line integral formula. Simplify the expression and integrate over the specified range of t values.

Q: What is the path equation used in the example?

The path equation used in the example is x(t) = 4t and y(t) = 3t.

Q: What is the line integral being calculated in the example?

The line integral being calculated in the example is the integral of the function f(x, y) = xy along the path C defined by x(t) = 4t and y(t) = 3t.

Summary & Key Takeaways

• The video explains how to evaluate a line integral along a given path using the formula: ∫f(x, y)√(x'^2 + y'^2) dt.

• The specific example in the video calculates the line integral of the function f(x, y) = xy along the path C defined by x(t) = 4t and y(t) = 3t.

• By plugging the path equations into the line integral formula and simplifying, the resulting integral is ∫0^112t^2 dt, which evaluates to 20.