What is the volume obtained by rotating the region enclosed by #y=11-x#, #y=3x+7#, and #x=0# about the y-axis?

Answer 1

#V= V_1-V_2=11,525pi/27#

Given: #y=11-x#, #y=3x+7#, and #x=0# Required: The volume obtained by rotating about the y-axis Solution strategy: looking at the figure we see that
#V= 2pi [int_(1)^11x(11-x)dx- int_(-2 1/3)^1x(3x+7)dx]#
#V_1 = 2pi [(11/2x^2-x^3/3)_1^11]# #V_2 = 2pi [(x^3+7x^2/2)_(-2 1/3)^1]# #V= V_1-V_2=11,525pi/27#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the volume obtained by rotating the region enclosed by ( y = 11 - x ), ( y = 3x + 7 ), and ( x = 0 ) about the y-axis, you can use the method of cylindrical shells.

The formula for the volume using cylindrical shells is:

[ V = 2\pi \int_{a}^{b} x \cdot f(x) , dx ]

Where:

  • ( f(x) ) is the height of the shell (the distance between the curves)
  • ( a ) and ( b ) are the x-values where the region starts and ends.

First, find the points of intersection of the curves: [ 11 - x = 3x + 7 ] [ x = 2 ]

Now, integrate from ( x = 0 ) to ( x = 2 ):

[ V = 2\pi \int_{0}^{2} x \cdot [(11 - x) - (3x + 7)] , dx ]

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7