Volume with parametric equations: x = 4 cos(theta), y = 2sin(theta); sketch from 0 to pi/2, and find...

[Aminta_1900]

"The parametric equations of a curve are x = 4 cos theta and y = 2 sin theta. Sketch the curve for values of theta from 0 to pi/2. Find the area in the first quadrant bounded by the curve and the axes"

I'm not sure how to go about this but began tentatively with


and tried to integrate. But since I've so far not learned how to differentiate inverse trig functions I assume this is not the method it is asking.

 

[blamocur]

I've so far not learned how to differentiate inverse trig functions

This is not difficult to find. You can use, among others, this Wikipedia page. But you don't have to if you don't want. As an alternative way, you can express [imath]dx[/imath] through [imath]d\phi[/imath] and [imath]\phi[/imath] and integrate over [imath]\phi[/imath].

 

[khansaheb]

"The parametric equations of a curve are x = 4 cos theta and y = 2 sin theta. Sketch the curve for values of theta from 0 to pi/2. Find the area in the first quadrant bounded by the curve and the axes"

I'm not sure how to go about this but began tentatively with

View attachment 36731
and tried to integrate. But since I've so far not learned how to differentiate inverse trig functions I assume this is not the method it is asking.

What do you get if you plot

(x/4)^2 + (y/2)^2 = 1 ?​

Now think.....

 

[khansaheb]

"The parametric equations of a curve are x = 4 cos theta and y = 2 sin theta. Sketch the curve for values of theta from 0 to pi/2. Find the area in the first quadrant bounded by the curve and the axes"

Did you solve it ?

 

[Dr.Peterson]

"The parametric equations of a curve are x = 4 cos theta and y = 2 sin theta. Sketch the curve for values of theta from 0 to pi/2. Find the area in the first quadrant bounded by the curve and the axes"

I'm not sure how to go about this but began tentatively with

View attachment 36731
and tried to integrate. But since I've so far not learned how to differentiate inverse trig functions I assume this is not the method it is asking.

Rather than focus on what you don't know, please tell us what you have learned about area in the context of parametric equations. Have you sketched the curve as requested? Were you given a formula for integrating such a region, like this or this?

 

[Harry_the_cat]

I know this is posted in the "calculus" section, but you really don't need calculus to find the required area.

 

[khansaheb]

The parametric equations of a curve are x = 4 cos theta and y = 2 sin theta.

When, x = ±4, what are the values of Θ ?

When, y = ±2, what are the values of Θ ?

Does "that" give you any insight regarding the shape of the curve?

 

[Dr.Peterson]

I know this is posted in the "calculus" section, but you really don't need calculus to find the required area.

When, x = ±4, what are the values of Θ ?

When, y = ±2, what are the values of Θ ?

Does "that" give you any insight regarding the shape of the curve?

Of course, all this requires either knowing enough about parametric equations to determine that the curve is an ellipse, or just guessing that based on a few points, and being confident enough to assume that. The big question is, what does @Aminta_1900 actually know about the subject? That's why I asked.

My impression is that the problem is intended to use something that has just been taught in calculus, and that parametric equations are a not entirely familiar idea.

 

[khansaheb]

Of course, all this requires either knowing enough about parametric equations to determine that the curve is an ellipse, or just guessing that based on a few points, and being confident enough to assume that. The big question is, what does @Aminta_1900 actually know about the subject? That's why I asked.

My impression is that the problem is intended to use something that has just been taught in calculus, and that parametric equations are a not entirely familiar idea.

he problem is intended to use something that has just been taught in calculus, and that parametric equations are a not entirely familiar idea.

I agree - that is why I am providing hints - one line at a time

 

[Aminta_1900]

I agree - that is why I am providing hints - one line at a time

Yes, I've solved this, thanks for the help. My PC would not log onto the site for a short time, so I couldn't reply.

I have it that these equations form an ellipse, and so the area asked would be one quarter of pi * 4 * 2 = 2pi units^2.

I hadn't been practicing parametric equations with trig, though the answer was right at the top of my trig notes.

I've posted this in calculus because initially I thought it involved integration, not having noticed the ellipse, until Khansaheb pointed it out.

 

[Dr.Peterson]

I have it that these equations form an ellipse, and so the area asked would be one quarter of pi * 4 * 2 = 2pi units^2.

I hadn't been practicing parametric equations with trig, though the answer was right at the top of my trig notes.

I've posted this in calculus because initially I thought it involved integration, not having noticed the ellipse, until Khansaheb pointed it out.

So it doesn't come from a calculus course, in which you have been taught about finding areas by integration? The current topic is trigonometry, or converting parametric equations into other forms? It's helpful when you tell us the context, so we can know what to recommend.

And you aren't interested in trying integration in parametric form (as in the links I gave), in order to learn them?

My PC would not log onto the site for a short time, so I couldn't reply.

That happens to me from time to time, and it takes (I think exactly) an hour to come back.

(In fact, coincidentally, it just happened. I hit Post, and wasn't able to connect again for exactly one hour.)