I put a few steps together rather than showing my work. When you complete the first integration, you should end up with y*sqrt(4x^2 + 5). Then using those limits of integration from 0 to x for y, this substitution gives x*sqrt(4x^2 + 5).
You could and keep the value as u^3/2 from 5 - 9, or keep the expression in terms of X and keep the bounds 0 - 1. You will ultimately end up with the same thing.
I use Stewart's Multivariable Calculus (8th edition) :) I have previously used Anton-Bivens-Davis Multivariable Calculus, which is a great resource as well.
your examples are excellent. i have a question to ask though. i don't know how to present a point in the three coordinate planes. so when you draw the figure for z=9 i get confused. i know i need to know how to draw the objects to find the interval of r as well as theta angle - yet i have no clue to draw when it relates to z-plane. can you please give me some sources (youtuve video, pdf files,...) that teach how to do that? i have searched in the internet ever since last year but still don't get the answer. since you draw it so smoothly i would like to ask you for a favour. thank you
Hello! Here is a video where I teach graphing quadric surfaces: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-HT32icHGl6k.html. This may help with graphing the complicated figures. Good luck!
Is it possible to analytically solve the surface area of a elliptic paraboloid? An example would be z=x^2+4y^2, or does a have to equal b in the formula for the shape to be solved by hand.
I dont understand how you got 4pi/3 in the end. After integrating 2 dtheta from 0 to 1, I got 2. Then integrating that I ended up with 8. What am I doing wrong?
We need to change the limits of integration - the 0 to 1 are the limits for x, not theta. Substituting 0 and 1 into x = 2sin(theta) and solving for theta, we get theta = 0 and pi/6. The outer limits of integration are okay, but the inner should be 0 to pi/6. I apologize for the poor notation!
point 0,0 to 1,1. you can do slope formula to get m and just y-y1 = m(x-x1) to get the formula, but it's pretty obvious because it goes from 0,0 to 1,1 and is a straight line that the equation is y=x
Are you referring to example 2? Around 8:40? Assuming so, I switched from rectangular to polar coordinates. The inner limits of integration refer to the radius, r, of the circle. The upper circle is x^2 + y^2 = 9, so the radius ranges from 0 (at the origin) to 3 (at z = 9).
