QuestionQuestion 13
Let R be the region bounded by the line and the curve .
Let S be the solid obtained by rotating R around the y-axis.
(a) Compute the area of R.
(b) Determine the volume of S. Specify the method you used (disk, washer, or shell).
Include the area, volume, and method in the text entry, and show all other work on your scratch paper.
Studdy Solution
STEP 1
What is this asking?
We need to find the area of a region defined by a curve and the x-axis, and then find the volume of the solid created when we spin that region around the y-axis.
Watch out!
Rotating around the y-axis can be tricky!
Make sure to choose the right integration method and set up your integral carefully.
STEP 2
1. Find the x-intercepts
2. Calculate the area
3. Choose a volume method
4. Set up and compute the integral for volume
STEP 3
To find where the curve intersects , we **set** equal to **zero**: .
STEP 4
This gives us and .
These are the **bounds of integration** for the area calculation!
STEP 5
The area of R is given by the **definite integral** of the curve from to .
STEP 6
STEP 7
**Expand** the integrand:
STEP 8
Now, we **integrate**:
STEP 9
**Evaluate** at the bounds:
STEP 10
So, the **area of R** is .
STEP 11
Since we're rotating around the y-axis, the **shell method** is a good choice.
Using the shell method avoids the need to express in terms of .
STEP 12
The **shell method formula** is , where is the height of the shell, and is the radius.
STEP 13
In our case, , and our **bounds** are and .
STEP 14
So, our integral is:
STEP 15
**Expand** the integrand:
STEP 16
**Integrate**:
STEP 17
**Evaluate** at the bounds:
STEP 18
The area of R is .
The volume of S is , calculated using the shell method.
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