Question

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.

Answer #1

Find the surface area of the solid generated when the
region bounded by x=ln(2y+1),0≤y≤1 is revolved about the Y-
axis.

4. Find the area of the region bounded by the curve y = 4 − x 2
and the line y = x + 2.

1- Find the area enclosed by the given curves.
Find the area of the region in the first quadrant bounded on the
left by the y-axis, below by the line above left
by y = x + 4, and above right by y = - x 2 + 10.
2- Find the area enclosed by the given curves.
Find the area of the "triangular" region in the first quadrant that
is bounded above by the curve , below by the curve y...

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.

Calculate the area, in square units, of the region bounded by
the line x=2 on the left, the curve f(x)=ln(x-6)+1 on the right,
the line y=3 above, and the x-axis below. Give an exact answer, in
terms of e.

Consider the region bounded by y=sqrt(x) and y=x^3
a) Find the area of this region
b) Find the volume of the solid generated by rotating this
region about the x-axis using washer
c) Find the volume of the solid generated by rotating this
region about the horizontal line y=3 using shells

Find the area of the region in the ?? x y -plane bounded above
by the graph of the function ?(?)=9 , below by the ? -axis, on the
left by the line ?=8 , and on the right by the line ?=19 . The area
is

a.) Find the length of the curve y=ln(x),1 ≤ x ≤ sqrt(3)
b.) Using disks or washers, find the volume of the solid
obtained by rotating the region bounded by the curves y^2=x and
x=2y about the y-axis
c.) Find the volume of the solid that results when the region
bounded by x=y^2 and x=y+12 is revolved about the y-axis

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.)

Estimate the area of the region bounded between the curve f(x) =
1 x+1 and the horizontal axis over the interval [1, 5] using a
right Riemann sum. Use n = 4 rectangles first, then repeat using n
= 8 rectangles. The exact area under the curve over [1, 5] is ln(3)
≈ 1.0986. Which of your estimates is closer to the true value?

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