(a) The accompanying figure shows a surface of revolution that is generated by revolving the curve y = f x in the xy -plane about the x -axis. Show that the equation of this surface is y 2 + z 2 = [ f x ] 2 . (b) Find an equation of the surface of revolution that is generated by revolving the curve y = e x in the xy -plane about the x -axis. (c) Show that the ellipsoid 3 x 2 + 4 y 2 + 4 z 2 = 16 is a surface of revolution about the x-axis by finding a curve y = f x in the xy -plane that generates it.
(a) The accompanying figure shows a surface of revolution that is generated by revolving the curve y = f x in the xy -plane about the x -axis. Show that the equation of this surface is y 2 + z 2 = [ f x ] 2 . (b) Find an equation of the surface of revolution that is generated by revolving the curve y = e x in the xy -plane about the x -axis. (c) Show that the ellipsoid 3 x 2 + 4 y 2 + 4 z 2 = 16 is a surface of revolution about the x-axis by finding a curve y = f x in the xy -plane that generates it.
(a) The accompanying figure shows a surface of revolution that is generated by revolving the curve
y
=
f
x
in the xy-plane about the x-axis. Show that the equation of this surface is
y
2
+
z
2
=
[
f
x
]
2
.
(b) Find an equation of the surface of revolution that is generated by revolving the curve
y
=
e
x
in the xy-plane about the x-axis.
(c) Show that the ellipsoid
3
x
2
+
4
y
2
+
4
z
2
=
16
is a surface of revolution about the x-axis by finding a curve
y
=
f
x
in the xy-plane that generates it.
The curve in the plane defined by the equation (*) (x^2 + xy + y^2 = 7) is a rotated ellipse, shown in the figure.
(c) The curve has two horizontal tangent lines. Find their equations.(d) Why can we see from the figure that the curve is not the graph of a function y of x?Show directly from (∗) that y is not a function of x.
The curve in the plane defined by the equation (*) (x^2 + xy + y^2 = 7) is a rotated ellipse, shown in the figure.
a) Use implicit derivation to find dy/dx expressed by x and y.
(b) Show that the curve passes through the point (2,1) and find the equation of the tangent line to the curve at the point (2,1).
University Calculus: Early Transcendentals (4th Edition)
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Area Between The Curve Problem No 1 - Applications Of Definite Integration - Diploma Maths II; Author: Ekeeda;https://www.youtube.com/watch?v=q3ZU0GnGaxA;License: Standard YouTube License, CC-BY