Use washers to find the volume formed by rotating the region enclosed by:
y=1.4−2|x−13| and y=0 about the y-axis
y1 = 1.4 -2(x-13) , y2 = 1.4 -2(-x+13)
y1 = 27.4 - 2x , y2 = -24.6 + 2x
intersection:
2x - 24.6 = -2x + 27.4
4x = 52
x = 13 , then y = 1.4 ---> range is 0 to 1.4
y1 = 27.4 - 2x ---> x = 13.4 - y/2 --> outer radius
y2 = -24.6 + 2x --> x = 12.3 + y/2 --> inner radius
Volume
= ?? ( (13.4 - y/2)^2 - (12.3 + y/2)^2 ) dy from 0 to 1.4
= ?? (28.27 - 25.7y ) dy from 0 to 1.4
= ?[28.27y - 12.85y^2 ] from 0 to 1.4
= ?( 39.578 - 25.195 - 0)
= ?(14.392)
= appr 45.214 units^3
verification:
http://www.wolframalpha.com/input/?i=%CF%80%E2%88%AB+(+(13.4+-+y%2F2)%5E2+-+(12.3+%2B+y%2F2)%5E2+)+dy+from+0+to+1.4
Well, if we're talking about washers, I hope they're going to help us clean up this problem. Let's get started!
To find the volume formed by rotating the region enclosed by these curves about the y-axis, we can use the method of washers. We'll be integrating with respect to y, so hold on tight!
First, let's figure out the limits of integration. We need to find the values of y where the two curves intersect. To do that, we set them equal to each other:
1.4 - 2|x - 13| = 0
Now, let's break that down into two separate equations:
1.4 - 2(x - 13) = 0 (when x - 13 is positive)
1.4 + 2(x - 13) = 0 (when x - 13 is negative)
By solving these equations, we find that x = 14.4 and x = 11.6. Note that we only consider the positive value for x, as we're looking for the volume enclosed by the curves.
Now, let's express the curves in terms of y. The equation y = 1.4 - 2|x - 13| can be rewritten as:
y = 1.4 - 2|13 - x|
Since we want to integrate with respect to y, we need to solve for x in terms of y:
2|13 - x| = 1.4 - y
|13 - x| = (1.4 - y)/2
Now, let's consider two cases:
Case 1: When x > 13
13 - x = (1.4 - y)/2
Solving for x, we get x = 13 - (1.4 - y)/2
Case 2: When x < 13
x - 13 = (1.4 - y)/2
Solving for x, we get x = 13 + (1.4 - y)/2
Now, let's move on to the integration part! The volume of each washer can be expressed as:
V = π[R^2 - r^2]dy
Where R is the outer radius (distance from the y-axis to the outer curve) and r is the inner radius (distance from the y-axis to the inner curve).
For this problem, the outer curve is y = 1.4 - 2|13 - x|, and the inner curve is y = 0.
So, the outer radius (R) is 13 - (1.4 - y)/2, and the inner radius (r) is 13 + (1.4 - y)/2.
Now, we can set up the integral:
V = ∫[13 - (1.4 - y)/2]^2 - [13 + (1.4 - y)/2]^2 dy
Now, let us integrate and evaluate the result using these limits of integration: y = 0 to y = 1.4.
And voila! You now have the volume formed by rotating the region enclosed by those curves about the y-axis using the method of washers. I hope this cleaned up any uncertainties!
To find the volume formed by rotating the region enclosed by the given curves about the y-axis, we can use the method of washers. Here's how to do it step-by-step:
Step 1: Find the boundaries of integration.
To determine the boundaries of integration, we need to find the x-values where the two curves intersect. Setting the two equations equal to each other, we get:
1.4 - 2|x - 13| = 0
Now we can solve this equation for x:
2|x - 13| = 1.4
|x - 13| = 0.7
Since absolute value expressions have two cases, we solve for both positive and negative values:
Case 1: x - 13 = 0.7
x = 13 + 0.7
x = 13.7
Case 2: -(x - 13) = 0.7
- x + 13 = 0.7
- x = 0.7 - 13
x = 12.3
Therefore, the boundaries of integration are x = 12.3 and x = 13.7.
Step 2: Express the curves in terms of y.
We need to rewrite the equations in terms of y in order to set up the integral properly. From the given equations:
y = 1.4 - 2|x - 13|
Let's split this equation into two parts based on the two cases for the absolute value, when x < 13 and x > 13:
When x < 13:
y = 1.4 - 2(13 - x)
y = 1.4 - 26 + 2x
y = 2x - 24.6
When x > 13:
y = 1.4 - 2(x - 13)
y = 1.4 - 2x + 26
y = -2x + 27.4
So, the equations in terms of y are y = 2x - 24.6 and y = -2x + 27.4.
Step 3: Set up the integral.
To find the volume, we integrate the area of each washer formed by rotating the region enclosed by the curves about the y-axis.
V = ∫[a,b] π(R^2 - r^2) dy
Where [a, b] represents the boundaries of integration. In this case, [a, b] = [0, ymax], and ymax is the maximum value for y in the region.
The outer radius (R) and the inner radius (r) are given by the equations of the curves.
Outer radius (R) = 2x - 0
Inner radius (r) = -2x + 0
Step 4: Evaluate the integral.
Putting it all together, the integral becomes:
V = ∫[0, ymax] π((2x)^2 - (-2x)^2) dy
Now we need to express x in terms of y to solve for ymax:
2x - 24.6 = y
2x = y + 24.6
x = (y + 24.6)/2
-2x + 27.4 = y
-2x = y - 27.4
x = (27.4 - y)/2
Since we are rotating about the y-axis, ymax will be the maximum value of y.
Step 5: Calculate the volume.
Evaluate the integral by plugging in the expressions for the radii and ymax:
V = ∫[0, ymax] π(((y + 24.6)/2)^2 - ((27.4 - y)/2)^2) dy
Now integrate this expression to find the volume:
V = π * ∫[0, ymax] (((y + 24.6)/2)^2 - ((27.4 - y)/2)^2) dy
Calculating this integral will give you the volume formed by rotating the region enclosed by the given curves about the y-axis.
To find the volume formed by rotating the region enclosed by the curves y = 1.4 − 2|x − 13| and y = 0 about the y-axis using washers, you need to follow these steps:
1. First, find the intersection points of the two curves. Set them equal to each other and solve for x:
1.4 − 2|x − 13| = 0
Since the absolute value is involved, you need to consider two cases:
Case 1: (x − 13) ≥ 0
1.4 − 2(x − 13) = 0
1.4 − 2x + 26 = 0
-2x = -27.4
x = 13.7
Case 2: (x − 13) < 0
1.4 − 2(-x + 13) = 0
1.4 + 2x − 26 = 0
2x = 24.6
x = 12.3
So, the intersection points are x = 12.3 and x = 13.7.
2. Next, integrate to find the volume of each washer. The washer method formula for finding the volume is:
V = π∫(outer radius^2 - inner radius^2) dy
Notice that the two curves bound the region, so the outer radius will be y = 1.4 − 2|x − 13| and the inner radius will be y = 0.
3. We're revolving the region about the y-axis, so express x in terms of y for the curves:
y = 1.4 − 2|x − 13|
1.4 − 2|x − 13| = y
|x − 13| = (1.4 − y) / 2
|x − 13| = (2.8 − 2y) / 4
Since we're revolving about the y-axis, take the absolute value away and solve for x:
x = 13 ± (2.8 − 2y) / 4
4. Determine the limits of integration. From the graph, you can see that the region is bounded between y = 0 and y = 1.4/2 = 0.7.
5. Finally, integrate the volume formula with respect to y over the given limits of integration:
V = π∫[0 to 0.7] [(13 + (2.8 − 2y) / 4)^2 - (13 - (2.8 − 2y) / 4)^2] dy
Evaluating this integral will give you the volume formed by rotating the region enclosed by the given curves about the y-axis using washers.