Question

Consider a lamina of density ρ = 1 the region bounded by the
x-axis, the

y-axis, the line x = 5, and the curve y = e^x.

Find the centroid ( ̄x, ̄y). (Simplify, but

leave your answer as an exact expression not involving integrals.)

Answer #1

Hope this will help you!

Consider the lamina of uniform density ρ bounded by y = 4, y =
x2, 0 ≤ x ≤ 2. Find the moment about the y-axis My.

A lamina with constant density ρ(x,
y) = ρ occupies the given region. Find the
moments of inertia Ix and
Iy and the radii of gyration x and
y.
The region under the curve y = 4 sin(x) from
x = 0 to x = π.

find the centroid of the lamina of occupying the region bounded
by y=x,y=9

find moment of inertia about x-axis of lamina with shape of the
region bounded by y=x^2 y=0 and x=0 with density=x+y

A lamina with constant density ρ(x, y) = ρ occupies the given
region. Find the moments of inertia Ix and Iy and the radii of
gyration x double bar and y double bar. The rectangle 0 ≤ x ≤ 2b, 0
≤ y ≤ 5h.
Ix
=
Iy
=
=
=

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.

5. Find the mass of the lamina which is bounded by y = 9 − x^2
and x-axis if ρ(x, y) = y.
6. Find the center of mass for the lamina in question 5.

Consider the region bounded by ? = 4? , ? = 1 and x-axis. Set
up the appropriate integrals
for finding the volumes of revolution using the specified
method and rotating about the specified axis. Be sure to first
sketch the region and draw a typical cross section. SET UP THE
INTEGRALS ONLY. DO NOT evaluate the integral.
a) Disc/washer method about the x-axis
b) Shell method about the y-axis
c) Disc/washer method about the line ? = 2.
d)...

Consider the region bounded by y = x2, y = 1, and the y-axis,
for x ≥ 0. Find the volume of the solid. The solid obtained by
rotating the region around the y-axis.

Find the area of the region bounded by the y-axis, the curve y =
ln(x+ 1),
and the tangent line to y = ln(x + 1) at x = 3.

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