The simplified expression is and the restriction is
The expression is given as:
Express x^2 - y^2 as (x + y)(x - y) and factorize other expressions
Rewrite the expression as products
Cancel out the common factors
Express 4x^2 - y^2 as (2x - y)(2x + y)
Cancel out the common factors
Take the LCM
Hence, the simplified expression is and the restriction is
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Answer:
The volume of the solid is
Step-by-step explanation:
In this case, the washer method seems to be easier and thus, it is the one I will use.
Since the rotation is around the y-axis we need to change de dependency of our variables to have . Thus, our functions with
as independent variable are:
For the washer method, we need to find the area function, which is given by:
By taking a look at the plot I attached, one can easily see that for a rotation around the y-axis the outer radius is given by the function and the inner one by
. Thus, the area function is:
Now we just need to integrate. The integration limits are easy to find by just solving the equation , which has two solutions
and
. These are then, our integration limits.
