How do I find the area of the larger region?

[Kulla_9289]

Hello,
I have an assignment and I am stuck at this last question. I do not understand the question by itself, in particular the last one (area), as my answer doesn't even remotely match their one. I need help understanding what the question wants and how would I achieve this. I know up to Calculus 2 except hyperbolic functions.

Thanks

 

[Dr.Peterson]

I do not understand the question by itself, in particular the last one (area), as my answer doesn't even remotely match their one. I need help understanding what the question wants and how would I achieve this.

Here is a graph (for a=1):

​

They want the area of overlap between the two colors.

They told you the location of the intersection (by just giving the angle a name); choose your limits of integration accordingly. You'll have two separate integrals.

 

[Dr.Peterson]

I just noticed that your title asks specifically about the larger region. You'll find that by subtraction.

 

[logistic_guy]

I do not understand the question by itself, in particular the last one (area)

Here is a brief explanation of each part.

First, to find the area of a polar curve we use this formula:

$\displaystyle A = \int_{a}^{b}\frac{1}{2} r^2(\theta) \ d\theta$

$\displaystyle \bold(a)$ Find the greatest distance of $\displaystyle r = a(\cos \theta + \sin \theta)$ from the pole (origin).

Take the derivative of $\displaystyle r$ and set it to zero. Once you get the angle $\displaystyle \theta + n\pi, \ n\in \mathbb{Z}$, check it for angles that are within the domain. If you get the diameter of the circle as the longest distance, then your calculations are correct.


$\displaystyle \bold(b)$ Find the small area enclosed by the two curves.

Solve $\displaystyle \int_{C_2} \cdots = \int_{0}^{\phi} \frac{1}{2} r^2 \ d\theta$
and $\displaystyle \int_{C_1} \cdots = \int_{\phi}^{3\pi/4} \frac{1}{2} r^2 \ d\theta$

Then add them together.


$\displaystyle \bold(c)$ Find the large area enclosed by the two curves.

Solve $\displaystyle \int_{C_1} \cdots = \int_{-\pi/4}^{3\pi/4} \frac{1}{2} r^2 \ d\theta$

Then subtract from it the area you found in part $\displaystyle \bold(b)$.


Note: The graph that was given by professor Dave will help you a lot to visualize what area you are looking for.

 

[Kulla_9289]

Is it possible to get the area without sketching the graph? Is it possible? My sketching skills aren't up to par

 

[logistic_guy]

Is it possible to get the area without sketching the graph? Is it possible? My sketching skills aren't up to par

Yes, it is.

Once you have the domain of one curve and the intersection points of the two curves, you can do it without sketching. But errors most likely will happen if the polar curves pass through the origin more than two times. Since the polar curves are very tricky, I advise you to learn how to sketch them or at least approximate their graph. The first lesson in polar curves is to teach you how to make a table and graph them. At least learn how to graph the basic ones.

In the test the polar curves usually will be given. If not, they will give you a very basic one (like a circle or an apple) that you can visualize easily.

 

[Dr.Peterson]

Is it possible to get the area without sketching the graph? Is it possible? My sketching skills aren't up to par

You don't need a very accurate sketch. Knowing the general shape, and plotting couple points (such as the y-intercept) should be enough. Again, you don't need to figure out the location of the intersection, which they have given as $\phi$. The important thing is just to know that there is only one such point.

 

[Kulla_9289]

I think I understand now. Thank you for the help