Question

there are 5 parts to this question

1.Find the area of the region R enclosed by the graphs of y = x 2 , the y-axis and the line y = 4

2.Find the volume of the solid generated by revolving the region in problem 1 about the x-axis.

3.Use the cylindrical shell method to find the volume of the solid generated by revolving the region in problem 1 about the y-axis.

4.Find the volume of the solid generated by revolving the region in problem 1 about the line x = −5

5.Find the volume of the solid generated by revolving the region in problem 1 about the line y = −2.

6.Find the volume of the solid whose base is the region in problem 1 and whose cross sections, perpendicular to the y-axis, are squares

Answer #1

Consider the region S enclosed by the graphs of y=x^3-6^2+9x and
y=x/2 . Determine which solid has the greater volume, and by how
much: (a) The solid generated by revolving S about the x-axis; (b)
The solid generated by revolving S about the line y=4.

Find the volume of the solid generated by revolving the region
bounded by the graphs of y = e x/4 , y = 0, x = 0, and x = 6 about
the x−axis.
Find the volume of the solid generated by revolving the region
bounded by the graphs of y = √ 2x − 5, y = 0, and x = 4 about the
y−axis.

Find the volume of the solid whose base is rotating around the
region in the first quadrant bounded by y = x^5 and y = 1.
A) and the y-axis around the x-axis?
B) and the y-axis around the y-axis?
C) and y-axis whose cross sections are perpendicular to x-axis
are squares

Consider the solid S described below. The base of S is the
region enclosed by the parabola y = 1 - 9x^2 and the x-axis.
Cross-sections perpendicular to the x-axis are isosceles triangles
with height equal to the base. Find the volume V of this solid.

1) Find the volume of the solid obtained by rotating the region
enclosed by the graphs about the given axis. ?=2?^(1/2), y=x about
y=6 (Use symbolic notation and fractions where needed.)
2) Find the volume of a solid obtained by rotating the region
enclosed by the graphs of ?=?^(−?), y=1−e^(−x), and x=0 about
y=4.5.
(Use symbolic notation and fractions where needed.)

Find the volume of the of the solid described as follows: The
base of the solid is the region enclosed by the line y=4-x, the
line y=x, and the y-axis. The cross sections of the region that are
perpendicular to the x-axis are isosceles triangles whose height is
equal to half their base. What is the volume of this solid (rounded
to two decimal places)? Please show work. Thanks much!

Find the volume of the solid generated by revolving the region
enclosed by the triangle with vertices(3,1),(3,5), and(6,5) about
the y-axis.

Find the volume of the solid generated by revolving the plane
region bounded by the graphs of y= \sqrt{x} , y=0, x=5, about the
line x=8.
Find the volume using: a) DISK/WASHER Method b) SHELL METHOD

Find the volume of the solid generated by revolving the plane
region bounded by the graphs of y= \sqrt{x} , y=0, x=5, about the
line x=8.
Find the volume using: a) DISK/WASHER Method b) SHELL METHOD

Problem (9). Let R be the region enclosed by y = 2x, the x-axis,
and x = 2. Draw the solid and set-up an integral (or a sum of
integrals) that computes the volume of the solid obtained by
rotating R about:
(a) the x-axis using disks/washers
(b) the x-axis using cylindrical shells
(c) the y-axis using disks/washer
(d) the y-axis using cylindrical shells
(e) the line x = 3 using disks/washers
(f) the line y = 4 using cylindrical...

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