# Quick help with line integrals

• I dun get it
In summary, the given integral involves a curve C consisting of two segments: one from (1,0,1) to (2,3,1) and another from (2,3,1) to (2,5,2). The solution involves finding the parametric equations for each segment and using them to calculate the integral. The final answer is 97/3, which differs from the book's answer of 97/3.
I dun get it

## Homework Statement

$$\int_{C}(x+yz)dx + 2xdy + xyzdz$$

C goes from (1,0,1) to (2,3,1) and (2,3,1) to (2,5,2)

## The Attempt at a Solution

For C going from (1,0,1) to (2,3,1)
$$x=1+t, y=3t, z=1; 0\leq t \leq 1$$
$$x'(t)=1, y'(t)=3, z'(t)=0$$

$$\int^{1}_{0}(1+t+3t)*1dt + 2(1+t)*3dt + 0$$
$$=\int^{1}_{0}1+4t+6+6tdt$$
$$[7t+5t^{2}]^{1}_{0}=12$$

For C going from (2,3,1) to (2,5,2)
$$x=2, y=1+2t, z=t; 1\leq t \leq 2$$
$$x'(t)=0, y'(t)=2, z'(t)=1$$

$$\int^{2}_{1}0 + 2*2*2dt + 2(1+2t)tdt$$
$$=\int^{2}_{1}8+2t+4t^{2}dt$$
$$[8t+t^{2}+\frac{4}{3}t^{3}]^{2}_{1}=\frac{55}{3}$$

Total C is $$12+\frac{55}{3} = \frac{91}{3}$$
However, the answer in the book says 97/3, so I'm not sure what I did wrong

Hi I dun get it!

Your 55/3 should be 8*1 + 1*3 + (4/3)*7 = 8 + 3 + 28/3 = 61/3

## 1. What is a line integral?

A line integral is a type of integral used in mathematics to calculate the area under a curve or the length of a curve in a multi-dimensional space. It is typically used to find the work done by a force along a path or the circulation of a vector field along a curve.

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

To calculate a line integral, first, you need to define the path or curve that you want to integrate over. Then, you need to find the parametric equations for the curve and determine the limits of integration. Next, you need to set up the integral using the appropriate formula for the specific type of line integral (i.e. work, circulation, etc.). Finally, you can evaluate the integral using techniques like substitution or integration by parts.

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

Line integrals have many applications in physics, engineering, and other scientific fields. For example, they are used to calculate the work done by a force along a curved path, the circulation of a fluid in a vector field, and the electric potential of a charged particle moving along a path. They are also used in the fields of fluid dynamics, electromagnetism, and thermodynamics.

## 4. Can line integrals be negative?

Yes, line integrals can be negative. This depends on the orientation of the curve and the direction of the vector field being integrated. If the curve and the vector field are aligned in opposite directions, the resulting line integral will be negative. If they are aligned in the same direction, the line integral will be positive. It is important to pay attention to the orientation when setting up and evaluating a line integral.

## 5. Are there any tools or software programs that can help with line integrals?

Yes, there are many tools and software programs available that can help with line integrals. Some popular options include Wolfram Alpha, MATLAB, and Mathematica. These programs can help with setting up and evaluating line integrals, as well as visualizing the curves and vector fields involved. Additionally, there are many online resources and tutorials available for learning and practicing line integrals.

Replies
2
Views
772
Replies
12
Views
1K
Replies
1
Views
941
Replies
2
Views
1K
Replies
2
Views
727
Replies
9
Views
477
Replies
8
Views
1K
Replies
3
Views
1K
Replies
3
Views
1K
Replies
4
Views
1K