How to find the volume of the solid obtained by rotating the region bounded by the given curves about the line x=6. x=y^4 , x=1 ?
See the explanation below.
I interpreted the question to include the region both above and below the
I've set it up to use washers, the representative slice is taken perpendicular to the axis of rotation and is shown in black. The dashed red lines above and below the slice show the greater and lesser radii of the washer. Call the
The thickness of the slice is
The volume of the representative washer is
We see that
The volume of the resulting solid is
# = pi int_1^1 (1112y^4+y^8)dy# We can simplify the arithematic a bit be using symmetry of the region and evaluating from
#0# to#1# and doubling the result.
#V = 2pi int_0^1 (1112y^4+y^8)dy#
# = 2pi(392/45) = (784pi)/45 ~~ 54.734#
By signing up, you agree to our Terms of Service and Privacy Policy
To find the volume of the solid obtained by rotating the region bounded by the curves (x = y^4), (x = 1), and the line (x = 6) about the line (x = 6), you can use the disk method.

First, determine the points of intersection of the curves (x = y^4) and (x = 1). Set the equations equal to each other and solve for (y):
(y^4 = 1)
Solving for (y), we get (y = 1) and (y = 1). These are the limits of integration.

Next, express the volume element as a function of (y). The radius of each disk is the distance from the line (x = 6) to the curve (x = y^4), which is (6  y^4). So, the volume element is (\pi \cdot (\text{radius})^2 \cdot dy).

Integrate the volume element from (y = 1) to (y = 1) to obtain the total volume:
(V = \int_{1}^{1} \pi \cdot (6  y^4)^2 , dy)

Solve the integral to find the volume.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
 Consider the parametric equation #x= 10(cost+tsint)# and #y= 10(sinttcost)#, What is the length of the curve from #0# to #((3pi)/2)#?
 What is the slope of #f(t) = (t+1,t)# at #t =0#?
 How do you find the parametric equations for the intersection of the planes 2x+yz=3 and x+2y+z=3?
 What is the derivative of #f(t) = (t^2lnt, t^2sint ) #?
 How do you differentiate the following parametric equation: # (ttsin(t/2+pi/3), 2tcos(pi/2t/3))#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7