Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y=1/x^2, y=0, x=4, x=5; about y=−2
the shell method seems best
the corners of the cross section are
(4,0), (5,0), (5,.04), (4,.0625)
not needed for solution, but helps to visualize
the radius of the shell is x+2
volume = 2π * r * L * thickness
= 2π * (x + 2) * (1 / x^2) * dx
integration limits are 4 to 5
have at it
the problem with the above scenario is that the shells have thickness dy, since the region is rotated around a horizontal axis.
v = ∫[1/25,1/16] 2πrh dy
where r=y+2 and h=x-4=(1/√y - 4)
v = ∫[1/25,1/16] 2π(y+2)(1/√y - 4) dy = 4913π/120000
Hmmm. Let's check that using discs of thickness dx:
v = ∫[4,5] π(R^2-r^2) dx
where r=51/25 and R=y+2=1/x^2+2
v = ∫[4,5] π((1/x^2+2)^2-(51/25)^2) dx = 4913π/120000
oops...got my axes swapped
thanks, Cal
To find the volume of the solid obtained by rotating the region bounded by the curves about the specified axis, we can use the method of cylindrical shells. Here's how to calculate it step by step:
Step 1: Draw a diagram of the region bounded by the curves. In this case, we have the curves y = 1/x^2, y = 0, x = 4, and x = 5. The region is bounded between x = 4 and x = 5 and between the x-axis and the curve y = 1/x^2.
Step 2: Determine the height of each cylindrical shell. The height of each shell will be the difference between the y-coordinate of the upper curve (y = 1/x^2) and the y-coordinate of the lower curve (y = 0). In this case, the height of each shell will be y = 1/x^2 - 0, which simplifies to y = 1/x^2.
Step 3: Determine the radius of each cylindrical shell. The radius of each shell will be the distance between the axis of rotation (y = -2) and the x-coordinate of the region at that height. In this case, the radius of each shell will be 2 + x.
Step 4: Set up the integral to calculate the volume of the solid. The volume of each shell can be calculated as V = 2πrh, where r is the radius and h is the height. Since we are rotating about the line y = -2, the integral to find the total volume is given by:
V = ∫[a,b] 2π(2 + x)(1/x^2) dx
where [a,b] represents the interval of integration, which in this case is [4,5].
Step 5: Evaluate the integral to find the volume. Integrate the function 2π(2 + x)(1/x^2) with respect to x over the interval [4,5]. This will give you the volume of the solid.
(Volume = ∫[4,5] 2π(2 + x)(1/x^2) dx)
Evaluating this integral will give you the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.