Download 23.) If the region enclosed by the y-axis, the line y=2, and the curve y

23.) If the region enclosed by the y-axis, the line y=2, and the curve y=
is revolved about the
y-axis, the volume of the solid generated is…
To find where the graphs intersect set the two equations equal to each other but they have to
be in the same terms for instance both has to be y=something or both have to be x=something.
=2. Square both sides and you get the x=4 so the point of intersection is (4,2).
1.) Decide if you want to use washers or shells or which one is possible to use. Let’s use
shells though.
2.) Set up the integral.
LA= lateral area LA=2
h=2-y
but we are using dx so y has to be related to x. The equation y=
them so plug
Now h=2 -
relates
in for y.
.
=
3.) Take the anti-derivative.
2x =
and
=
dx
=
=
=
((
– ((
)
(16 - ) – (0-0)
=
=
which is A.
–Von Pursley
24.) The expression
(
…
)
is the width of the rectangles or the dx. Then look inside the parenthesis. Remember,
this is an approximation so
is close to zero so we know that we will be integrating starting
at 0. Then look at the final number (
) this is equal to 1 so now we know we are integrating
from 0 to 1. Remember that an integral is all the heights from a to b multiplied by dx.
is a
height and so are all the other numbers. The Riemann sum is an estimate for the infinite
number of heights there are between 0 and 1 such as 0.00000001, 0.00000002, 0.000000003
and so on. The radicals in the parenthesis is used to mock these infinite heights so the equation
is
. Also if you look at C when you plug in 1 for x you get
1 like we said it should be. So the answer is D
dx.
for the end value which is not
25.)
=
1. Take antiderivative integration by parts.
u
v
x
.5cos(2x)
1 dx
du
dv
sin(2x)dx
antiderivative = uv = x(-.5cos(2x)) –
= -.5xcos(2x) =
It is divided by 2 because when you take the
derivative of the top portion of the fraction, you have to take the derivative of the
outside function and the inside function. When you take the derivative of the outside
function you get the original equation (
) but then when you take the
derivative of the inside function (2x) you get 2. So you end up with (
).
The times 2 is not part of the original equation so we divide (-.5sin(2x)) by 2 to get the
original equation.
=-.5xcos(2x) -
+C
=
which is A.
2. You can also use tabular method.
DERIVATIVE
X
+
1
0
-
ANTIDERIVATIVE
sin(2x)
-
+C
Multiply through and you get the same thing.
-Von Pursley