Question

Let X be the region bounded by y=x and y=x^2.

Find the volume of the solid that is made when X is revolved
about the y-axis. Find it in two different methods, include a
diagram for each method, and explain.

Answer #1

Let R be the region bounded by the curves y = x, y = x+ 2, x =
0, and x = 4. Find the volume of the solid generated when R is
revolved about the x-axis. In addition, include a carefully labeled
sketch as well as a typical approximating disk/washer.

Let R be the region bounded by y = x2 + 1, y = 0, x =
1, and x = 2. Graph the region R. Find the
volume of the solid generated when R is revolved
about the y-axis using (a) the Washer Method and
(b) the Shell Method.

Let R be the region bounded by y=ln(x), the x-axis, and
the line x=e. Find the volume of the solid that is generated when
the region R is revolved about the x-axis.

Find the volume of the solid obtained by rotating the region
bounded by x = y^2 and x = |y| about the y-axis.?

Find the Volume of the solid obtained by revolving the region
bounded by y=-x^2+1, the x-axis, and the y-axis. Solve by using
shell method & another method.

A. For the region bounded by y = 4 − x2 and the x-axis, find
the volume of solid of revolution when the area is revolved
about:
(I) the x-axis,
(ii) the y-axis,
(iii) the line y = 4,
(iv) the line 3x + 2y − 10 = 0.
Use Second Theorem of Pappus.
B. Locate the centroid of the area of the region bounded by y
= 4 − x2 and the x-axis.

Find the volume of the solid generated by revolving the region
bounded by y = 2ex - 4x, y = 2 - 2x, x = 0, x = 1 about
the x-axis using the most appropriate method.

Find the volume V of the solid obtained by rotating the
region bounded by the given curves about the specified line.
y = 5x4, y = 5x, x ≥
0; about the x-axis
Find the area of the region enclosed by the given curves.
y = 3 cos(πx), y = 12x2 −
3
Find the volume V of the solid obtained by rotating the
region bounded by the given curves about the specified line.
2x = y2, x = 0, y =
5; about the...

Find the volume of the solid generated by revolving the region
bounded by y = sqrt(x) and the lines and about y=2 and x=0
about:
1) the x-axis.
2) the y-axis.
3) the line y=2.
4) the line x=4.

3. Find the volume of the solid of revolution. The region is
bounded by y= 4x and y = x^3 and x ≥ 0.
a) Make a sketch.
b) About the x axis (disk/washer method).
c) About the x axis (cylindrical shells).
d) About the y axis (disk/washer method).
e) About the y axis (cylindrical shells).

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