Solving Line Integral Problem on Curve z=x+y with Simple Parametric Equations

In summary, a simple line integral problem involves calculating the integral of a function along a straight line in a two-dimensional plane. The formula for calculating a simple line integral is ∫<sub>a</sub><sup>b</sup> f(x) dx, where a and b represent the starting and ending points of the line, and f(x) is the function being integrated. This problem can be solved by breaking the line into small segments and approximating the integral. Some real-world applications of simple line integrals include finding work done by a force, determining the center of mass, and calculating electric potential. Some common mistakes when solving these problems include not considering direction, not breaking the line into segments, and using the wrong formula. It is
  • #1
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Homework Statement



Evaluate the line integral,

[tex]\int_{C}x^{2}yzds, \text{ where C is the curve } z = x + y, \quad x + y + z = 1\text{ from }(1,\frac{-1}{2},\frac{1}{2}) \text{ to } (-3, \frac{7}{2}, \frac{1}{2})[/tex]

Homework Equations





The Attempt at a Solution



Here's my attempt at the problem,

attachment.php?attachmentid=31302&stc=1&d=1294959785.jpg


It lists the correct answer as, [tex]\frac{37\sqrt{2}}{3}[/tex] solved with different parametric equations and bounds for t as well.

What am I doing wrong?
 

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  • #2
I see one problem right away:
[tex]\sqrt{(1)^{2}+(-1)^{2}} \ne 1[/tex]
 

What is a simple line integral problem?

A simple line integral problem involves calculating the integral of a function along a straight line in a two-dimensional plane. It is a fundamental concept in calculus and is used to find the area under a curve or the work done by a force along a path.

What is the formula for calculating a simple line integral?

The formula for a simple line integral is ∫ab f(x) dx, where a and b represent the starting and ending points of the line, and f(x) is the function being integrated.

How is a simple line integral problem solved?

A simple line integral problem can be solved by breaking the line into small segments and approximating the integral by summing the areas of each segment. As the segments get smaller, the approximation becomes more accurate, and the solution approaches the true value of the integral.

What are some real-world applications of simple line integrals?

Simple line integrals have many practical applications, such as calculating the work done by a force on an object, finding the center of mass of a continuous distribution of mass, and determining the electric potential in a circuit.

What are some common mistakes made when solving simple line integral problems?

Some common mistakes when solving simple line integral problems include not considering the direction of the line, forgetting to break the line into small segments, and not using the correct formula for the specific problem. It is also essential to check for any discontinuities or singularities in the function being integrated.

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