What is the volume of a triangle bounded by y=-x+6, x=2, and y=2 rotated about the x-axis

Answer 1

y=x+6 has slope 1

and intersects x=2 at (2, 8) y=2+6

intersects y=2 at (-4,6) 2=-x+6

the triangle is from (-4,2) to (2,2) to (2,8)

It's a right triangle, base 6, height 6, area = 36/2 = 18

18xheight 8x2 = 18x16 = 288pi

try another approach

rotate over the x-axis

the highest point (2,8) has a circular path with area of the circle = 64pi

subtract the circular area from the path of (2,2) with area 4pi. 64-4 = 60pi

that's over a base of 6 (2--4 = 6) . 6x60pi = 360pi

That's an upper bound. the actual volume should be somewhat lower, about midpoint

between 360pi and 180pi, around 270pi

standard way to do this problem is take the integral of the outer radius squared - inner radius squared,

after factoring out pi

evaluated at between -x=-4 and x=2

square the outer radius (6-x)^2 and subtract the square of the inner radius 2^2 =4

36-12x + x^2 -4 = 32-12x + x^2

Its indefinite integral is 32x -12x^2/2 + x^3/3

= 32x -6x^2 + x^3/3

evaluated at x=-4

= 32(-4)-6(-4)^2 + (-4)^3/3

= -128 - 96 -64/3 = -224 - 64/3

evaluated at x=2

= 32(2) -6(2)^2 + (2)^3/3

=64 -24+ 8/3= 40 + 8/3

subtract the first from the second

40+ 8/3 + 224 + 64/3

= 264 + 72/3 = 264 + 24 = 288

the volume is 288pi

The calculations lead to frequent mistakes

try working it different ways and the closer they are the more likely you're right

Answer 2

I will use the washer method. Since we are rotating about the x axis, we don't need to swap x and y, so we can leave the formula as is:

V = π ∫(f(x)² – g(x)²)dx

In our case

f(x) = –x + 6

g(x) = 2

Integral bounds:

Since the shape is bound on the left by x=2, our left boundary condition on the integral is 2. For the right boundary, we need to solve where f(x) and g(x) intersect, so where f(x) = g(x)

–x + 6 = 2

x = 4 <-- this is our right bound.

Now we can evaluate the integral (from 2 to 4):

V = π ∫((–x + 6)² – 2²)dx

V = π ∫(x² – 12x + 36 – 2²)dx

V = π ∫(x² – 12x + 32)dx

V = π * (x³/3 – 12x²/2 + 32x)

V = π * (4³/3 – 12*4²/2 + 32*4 – (2³/3 – 12*2²/2 + 32*2))

V = 32π/3

Answer 3

desmos.com/calculator/b8k0czyiba

This video did not see the negative sign in front of the x for y = -x + 6. So the video answers a different question.

Updated graphL

desmos.com/calculator/kz2ij9k0dp