# Calculate the following line integral

• lep11
You asked if there are alternative ways, and the answer is yes. One alternative way is to use the fact that if ##g## and ##h## are continuous on ##[a,b]## and have continuous derivatives on ##(a,b)##, then ##\int_a^b{g'(t){\cdot}h(t){\cdot}dt}=g(b)h(b)-g(a)h(a)-\int_a^b{g(t){\cdot}h'(t){\cdot}dt}##.
lep11

## Homework Statement

Let ##f(x,y)=(xy,y)## and ##\gamma:[0,2\pi]\rightarrowℝ^2##,##\gamma(t)=(r\cos(t),r\sin(t))##, ##r>0##. Calculate ##\int_\gamma{f{\cdot}d\gamma}##.

## The Attempt at a Solution

The answer is 0. Here's my work. However, this method requires that you are familiar with some useful trig identities.

Could someone please take a look at it and check if it's correct? Are there alternative ways? I have also tried to find the potential function ##u##, ##\nabla{u}=f##...

Last edited:
lep11 said:

## Homework Statement

Let ##f(x,y)=(xy,y)## and ##\gamma:[0,2\pi]\rightarrowℝ^2##,##\gamma(t)=(r\cos(t),r\sin(t))##, ##r>0##. Calculate ##\int_\gamma{f{\cdot}d\gamma}##.

## The Attempt at a Solution

The answer is 0. Here's my work. However, this method requires that you are familiar with some useful trig identities.

Could someone please take a look at it and check if it's correct? Are there alternative ways? I have also tried to find the potential function ##u##, ##\nabla{u}=f##...

I get an answer of 0 as well, but I have not checked your work because I do not look at posted images, but only at typed versions.

## 1. What is a line integral?

A line integral is a mathematical tool used in multivariable calculus to calculate the total value of a function along a given curve or path.

## 2. How do you calculate a line integral?

To calculate a line integral, you first need to determine the parametric equation of the given curve or path. Then, you integrate the function along this curve with respect to its independent variable. The resulting value is the line integral.

## 3. What is the difference between a line integral and a regular integral?

A regular integral calculates the area under a curve in a two-dimensional plane, while a line integral calculates the value of a function along a given curve in a three-dimensional space.

## 4. What are some real-world applications of line integrals?

Line integrals are used in many fields of science and engineering, such as physics, electrical engineering, and fluid mechanics. They can be used to calculate work done by a force, electric and magnetic fields, and fluid flow along a given path.

## 5. What are the different types of line integrals?

There are two types of line integrals: the line integral of a scalar function and the line integral of a vector field. The former calculates the total value of a scalar function along a curve, while the latter calculates the work done by a vector field along a curve.

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