I having a hard time on solving ths question is there any way I can get help on it
So for a plastic bowl it can be seen as the function:
f(x) = x^12
and from x= 0 to x= 1 in conjunction of the graph:
g(x) = -x^20 + 2
with x = 0.9 to x = 1 and using these to move about the y axis
- how can we use integration to determine the volume?
-how can we utilize cylindrical shell intergration to determine the volume?
-Is there a surface integral that can we can use to find the surface area?
if you can help thanks!
the two curves intersect at (1,1) and (-1,1)
Since both curves are symmetric about the y-axis, the volume can be calculated integrating on the interval [0,1], using shells of thickness dx
v = ∫[0,1] 2πrh dx
where r = x and h = (2-x^2) - x^12
The surface is a bit trickier, since the two parts have to be done separately. The bottom would be
∫[0,1] 2πr ds
where r = x and ds = √(1+(12x^11)^2) dx
Thank you, so how I go about determining the cross sections?
Using shells, there are no cross-sections. That's the disc method, which doesn't work too well here, because you cannot express x as a function of y.
And besides, for a solid of revolution, all cross-sections are circles (or washers).
Sure, I can help you with that!
To determine the volume of the plastic bowl, you can use integration. Since the bowl is defined by the function f(x) = x^12, you need to integrate this function over the given range, from x = 0 to x = 1. The integral of f(x) will give you the volume between the graph and the x-axis.
The formula for finding the volume using integration is:
V = ∫[a,b] f(x) dx
In this case, you need to evaluate the integral:
V = ∫[0,1] x^12 dx
To integrate x^12, you can use the power rule for integration, which states that if a function is in the form x^n, the integral is (1/(n+1))x^(n+1). Applying the power rule to x^12, the integral becomes:
V = (1/13)x^(12+1) evaluated from 0 to 1
= (1/13)x^13 evaluated from 0 to 1
= (1/13)(1^13 - 0^13)
= 1/13
So, the volume of the plastic bowl is 1/13.
Now, let's move on to utilizing cylindrical shell integration to determine the volume. Cylindrical shell integration can be used to find the volume of a solid of revolution. Since the bowl is formed by rotating the graph of g(x) = -x^20 + 2 around the y-axis, you can use cylindrical shell integration to find its volume.
The formula for cylindrical shell volume is:
V = 2π∫[a,b] x * f(x) dx
In this case, you need to evaluate the integral:
V = 2π∫[0.9,1] x * (-x^20 + 2) dx
Simplify the integrand:
V = -2π∫[0.9,1] x^21 - 2x dx
To integrate x^21, use the power rule as before:
V = -2π * (1/22)x^(21+1) - (2/2)x^2 evaluated from 0.9 to 1
= -2π * (1/22)x^22 - x^2 evaluated from 0.9 to 1
Evaluating the integral, you can find the volume of the bowl using cylindrical shell integration.
Regarding surface area, a surface integral can be used to find the surface area of a three-dimensional object. However, in this case, since the plastic bowl is defined by two functions (f(x) and g(x)), it would require more specific information about the shape of the bowl to determine an appropriate surface integral.
I hope this explanation helps you understand how to use integration and cylindrical shell integration to determine the volume of a plastic bowl!