# Integration: Finding Area Between Curve along X and Y axis Questions

• Ahlahn
In summary: Question 2: In this question, you are given two curves and asked to find the area between them. You can do this by integrating the two curves, and taking into account that the lines will intersect at certain points. To find the points of intersection, you would set the two curves equal to each other and look for which one is greater. Question 3: This question asks you to find the area enclosed by two curves. To do this, you need to integrate the curves and take into account the points of intersection. Find these points by setting the two curves equal to each other and finding which is greater.
Ahlahn
Hello Everyone,
Below are three questions relating to the integration between curves that have been bugging me for some time now. I know how to integrate between curves. I know how to integrate between x and y, but when given equations like the ones below I am lost as to how to figure out 2 things

1. What the graphs look like(so I know which one is left/right, top/bottom)
2. The points of intersection(to figure out the interval [a,b])

You don’t have to answer all of my questions(Though that would be Awesome), but it would really help if you could point me in the right direction by telling me what I’m doing wrong so I won’t keep running into the same problems again. Thanks so much!

Question 1

Find the area of the region enclosed by the semicubical parabola y2 = x3 and the line x = 3
.

I am having a lot of trouble with questions like this one. I am assuming that the question is asking me to integrate along the y-axis between the curves x=y^2/3 and x=3, but what is the interval? How can I find the interval when given functions like these? What am I doing wrong?

Question 2

Sketch the region enclosed by the curves and compute its area as an integral along the x- or y-axis.
x = y3 - 18y, y + 7x = 0

I rewrote it as ...X = y^3 – 18y and x = -(1/7)y
Like the previous question, I have trouble figuring out how the graphs look like, which one lies on top or side- so I am clueless as to whether I should integrate along the x or y…

Question 3

Find the area enclosed by the curves listed below as a function of c.
y = c - x^2
y = x^2 - c

Okay so I plugged a random constant for c and found that the function y = c-x^2 lies on top, therefore we are integrating along the x axis. I then tried looking for the points of intersection, and failed. How can I find the interval if c is unknown?! O_O

1. Unfortunately, knowing what graphs look like is something that sort of just comes with experience- you need to know some of the basic ones like x^2, x^3, x^(1/2), 1/x, etc. and how variations on these affect the graph, like how you can graph (x-3)^2 by knowing what x^2 looks like. I learned these things in pre-calc. If you didn't learn these things, I would suggest that either you teach yourself or rely on your graphing calculator.

2. Figuring out the area you're dealing with, and the points of intersection, is easiest if you have a rough idea of what the graphs of your function look like. It's going to be harder for you if you don't know this.
Generally when a problem gives two curves, you can assume that they intersect at two points. Find these points of intersection by setting the functions equal to each other. To see which is above, pick any point in between the points of intersection and see which is greater. This method does not necessarily apply to area problems that specify more than two functions, though. You might have something that says "Find the area between f(x), g(x), x=a, x=b", in which case your limits of integration are a and b.

Question 1: Since this problem gives you two curves, you can first assume that they intersect at two points. So set them equal, $$y^{2/3}=3$$ and you will find your interval (taking into account that a square root will give you a positive and negative value). Alternatively, if you recognize that $$x=y^{2/3}$$ is symmetric about the x-axis, you can integrate from 0 to the positive square root value and multiply the integral by 2.
Also, you don't necessarily have to integrate along the y-axis. Again, if you recognize that it's symmetric about the x-axis, since $$y=0^{3/2}=0$$ you can integrate with respect to x from 0 to 3 and multiply the integral by 2.
I uploaded a picture to illustrate what I mean.

#### Attachments

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You also need to realize that if you integrate with respect to y in Question 1, your integral is going to be
$$\int\left(3 - y^{2/3}\right)dy$$
which is slightly more complicated than if you integrated with respect to x.

## What is integration?

Integration is a mathematical process that involves finding the area under a curve or between two curves. It is used to solve problems in calculus, physics, and engineering.

## Why is finding the area between curves important?

Finding the area between curves can help us calculate various quantities, such as volume, work, and displacement. It also allows us to analyze and understand the behavior of functions and their relationships.

## What is the difference between finding the area under a curve and between two curves?

Finding the area under a curve involves finding the area between a single curve and the x-axis, while finding the area between two curves involves finding the area between two different curves. This can be useful when comparing two different functions or determining the intersection points between them.

## What are the steps for finding the area between curves?

The steps for finding the area between curves are:

• 1. Find the points of intersection between the two curves.
• 2. Determine the boundaries for integration (x or y values).
• 3. Set up the integral using the appropriate integration formula.
• 4. Solve the integral to find the area between the curves.

## What are some real-life applications of finding the area between curves?

Finding the area between curves has many practical applications, such as calculating the volume of irregularly shaped objects, determining the displacement of an object with changing velocity, and finding the work done by a varying force. It is also used in fields like economics, biology, and chemistry to analyze and model data.

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