Area between Curves: Find Area Enclosed by y=x-1 and y^2=2x+6

In summary, the conversation discusses finding the area enclosed by the line y = x-1 and the parabola y^2 = 2x+6 in Example 6 of James Stewart Calculus Early Transcentals 6E. The question is about why the region between the graphs would need to be split into two pieces when integrating with respect to x. The answer clarifies that this is because the second equation has a symmetry along the x axis and the integral would need to be calculated between different bounds.
  • #1
samona
19
0

Homework Statement



Find the area enclosed by the line y = x-1 and the parabola y^2 = 2x+6



The Attempt at a Solution


This is Example 6 in Jame's Stewart Calculus Early Transcentals 6E. I'm trying to figure out why he states that if we were to integrate with respect to x instead of y, then we would have had to split up the region in two pieces.

I'd appreciate it if someone could help clarify that statement.

The link to the example is:

http://books.google.com/books?id=xU...QHNsrzYDw&ved=0CFwQ6AEwCQ#v=onepage&q&f=false
 
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  • #2
You would have to split the region between the graphs in two pieces mainly because the second equation (y2=2x+6), when solved for y, reveals a symmetry along the x axis. You would have to integrate (-(x-1)+(2x+6).5) between a and b (which you calculate) and 2(0-(-(2x+6).5)) between a and c. With c<b.
 
Last edited:
  • #3
Thank you! :smile:
 

1. How do I find the area enclosed by two curves?

To find the area enclosed by two curves, you first need to graph the curves and determine the points where they intersect. Then, you can use the definite integral to find the area between these two points. In this case, we are given the equations y=x-1 and y^2=2x+6, so we can graph these curves and find their points of intersection.

2. Can I use any method to find the area enclosed by two curves?

Yes, there are various methods that can be used to find the area enclosed by two curves. Some common methods include the definite integral, the method of disks or washers, and the method of shells. The method you choose may depend on the complexity of the curves and your personal preference.

3. Do I need to know calculus to find the area between curves?

Yes, to find the area between curves, you will need to use calculus concepts such as integration. The definite integral is used to find the area under a curve, so it is necessary to have a basic understanding of calculus to calculate the area enclosed by two curves.

4. Is there a specific formula for finding the area between two curves?

No, there is not a specific formula for finding the area between two curves. The area between curves is found by using the definite integral, which involves finding the antiderivative of a function and evaluating it at the limits of integration.

5. Can I use technology to help me find the area enclosed by two curves?

Yes, you can use technology such as a graphing calculator or a computer program to help you find the area enclosed by two curves. These tools can help you graph the curves, find the points of intersection, and evaluate the definite integral to find the area.

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