Can you integrate with respect to y to find the area bound between two curves?

[anonymity]

Find the area bound between y² = x + 5, and y²= 3 - x.I can't figure out how to put limits of integration, the integrand, or really anything besides just the integral sign to work with Latex, so bear with me (or better yet, direct me to a tutorial! I will search for one after this post, if there is one on the forum, don't bother mentioning it, i'll find it. If there is not, I would appreciate a shove in the right direction.)

First off, i solved the equations to get a function of y, which yields:

x = y² - 5, and x = 3 - y², and set up an integral with limits of integration of y = -2, and y = 2.

My makeshift integral will be represented by: "*|"
A = *| (y² - 5)² - (3 - y²)² = *| 34 - 16y² = 136

Is my answer correct? And more importantly, is this how I should be approaching a problem like this?

 

[Roni1985]

I think if you take the integral in terms of y, you will get the answer.

 

[dickdick]

Find the area bound between y² = x + 5, and y²= 3 - x.


I can't figure out how to put limits of integration, the integrand, or really anything besides just the integral sign to work with Latex, so bear with me (or better yet, direct me to a tutorial! I will search for one after this post, if there is one on the forum, don't bother mentioning it, i'll find it. If there is not, I would appreciate a shove in the right direction.)

First off, i solved the equations to get a function of y, which yields:

x = y² - 5, and x = 3 - y², and set up an integral with limits of integration of y = -2, and y = 2.

My makeshift integral will be represented by: "*|"
A = *| (y² - 5)² - (3 - y²)² = *| 34 - 16y² = 136

Is my answer correct? And more importantly, is this how I should be approaching a problem like this?

That's good start. But why are you squaring the values of x? Isn't the area just the integral (x_max-x_min)dy?

 

[anonymity]

That's good start. But why are you squaring the values of x? Isn't the area just the integral (x_max-x_min)dy?

Oh duh, yeah I don't know why i did that. I must have had volumes of revolution stuck in my head. I was differentiating with respect to y, sorry for leaving the "dy"s off.

I'm starting with a function of y because half of the graph is below 0, which means that I would have to set up multiple integrals. Differentiating with respect to y let's you go from top to bottom.

how's this:

A = *| (3-y²) - (y² - 5) dy = *| 8 dy = (8*2) - (8*-2) = 32

 

[dickdick]

Oh duh, yeah I don't know why i did that. I must have had volumes of revolution stuck in my head. I was differentiating with respect to y, sorry for leaving the "dy"s off.

I'm starting with "x"s because half of the graph is below 0, which means that I would have to set up multiple integrals. Differentiating with respect to y let's you go from top to bottom.

how's this:

A = *| (3-y²) - (y² - 5) dy = *| 8 dy = (8*2) - (8*-2) = 32

The y^2 don't cancel. Are you overtired?

 

[anonymity]

The y^2 don't cancel. Are you overtired?

Yes either that or just foolish.

Is the idea behind it right, differentiating with respect to y and using those limits of integration? That's what I'm mainly concerned with..taking the integral is child's play, though apparently not for me at the moment.

 

[dickdick]

Yes either that or just foolish.

Is the idea behind it right, differentiating with respect to y and using those limits of integration? That's what I'm mainly concerned with..taking the integral is child's play, though apparently not for me at the moment.

You aren't differentiating dy, you are just integrating (x_max-x_min)*dy between the two values of x where the curves intersect. It's pretty much the way you find an area. I wouldn't worry about your conceptual grasp on the problem. You just aren't getting the right answer.

 

[anonymity]

You aren't differentiating dy, you are just integrating (x_max-x_min)*dy between the two values of x where the curves intersect. It's pretty much the way you find an area. I wouldn't worry about your conceptual grasp on the problem. You just aren't getting the right answer.

I understand how to find the area between two curves work. My question was, if you spent the time to notice, was if it works if you use y values for limits of integration and integrate with respect to y. I found this out, and you can; thanks for (not) helping.

The answer was 64/3. If you had such a great and almighty conceptual grasp on the problem, why not just say this?