Set up a triple integral for the volume of the solid that lies
below the plane x + 2y + 4z = 8, above the xy-plane, and in the
first octant.
Hint: Try graphing the region and then projecting into the
xy-plane. To do this you need to know where the plane
x+ 2y + 4z = 8 intersects the xy-plane (i.e. where z = 0).
1-) Set up (but DO NOT COMPUTE) an integral for the volume of
the solid obtained by rotating the region bounded by the graphs of
y = 0, y = √ x − 2, and x = 4 around the y-axis.
2-) Find the area enclosed by one petal of the four-leaved rose
curve r(θ) = sin(2θ).
Instructions: For each solid described, set up, BUT DO
NOT EVALUATE, a single definite integral that represents the exact
volume of the solid. You must give explicit functions as your
integrands, and specify limits in each case. You do not need to
evaluate the resulting integral.
1. The solid generated by rotating the region enclosed by the
curves y = x^2 and y = x about the line x-axis.
Set up an integral that uses the disk method to find the volume
of the solid of revolution obtained by revolving the area between
the curves y = sech(x/2), y =2, x =0 and x = 4 around the line y=2.
Include a sketch of the region and show all work to integrate and.
Note: Recall that sech(u) = 1/cosh(u).
Please show details for every single step
set
up an integral to find the volume of the solid generated when the
region bounded by
y=x^2 and y=3x
i) rotate about x-axis using washer method
ii) Rotate about y-axis using washer method
iii) rotate abt y= -2 using the shell method
iv) rotatate about x=10 using the shell method
Set up an integral to find the volume of the solid generated
when the region bounded by y = x^3 and y = x^2 is (a) Rotated about
the x-axis using washers (b) Rotated about the y-axis using shells
(c) Rotated about the line y = −2 using either washers or
shells.
Use a triple integral to find the volume of the solid under the
surfacez = x^2 y and above the triangle in the xy-plane with
vertices (1.2) , (2,1) and (4, 0).
a) Sketch the 2D region of integration in the xy plane
b) find the limit of integration for x, y ,z
c) solve the integral
(sry abt this but, please read the question properly, i've
already recieved 3 wrong answers because the one who answered didnt
look the...
Draw the graph, solid of revolution, one representative disk/
washer.
Set up and evaluate the integral that gives the volume of the
solid formed by revolving the region formed by
a) when revolved about y-axis, the volume is ?
b) when revolved about x-axis, the volume is ?
c) when revolved about the line y=8, the volume is ?
d) when revolved about the line x=2, the volume is ?