The Region "R" under the graph of y = x^3 from x=0 to x=2 is rotated about the y-axis to form a solid.
a. Find the area of R.
b. Find the volume of the solid using vertical slices.
c. Find the first moment of area of R with respect to the y-axis. What do you notice about the integral?
d. Find the x coordinate of the centroid of R.
e. A theorem of Pappus states that the volume of a solid of revolution equals the area of the region being rotated times the distance the centroid of the region travels. Show that this problem confirms this theorem.
I was able to do part "a" as the integral from 0 to 2 of x^3 dx. Also I believe part "b" is pi*[3y^(5/3)/5] evaluated from 0 to 2
Assistance needed.
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Let's go through each part of the problem step by step:
a. To find the area of region R under the graph of y = x^3 from x = 0 to x = 2, we can integrate the function over the given interval. The integral represents the area between the curve and the x-axis.
The integral for the area can be set up as follows:
Area = ∫[0,2] x^3 dx
To evaluate this integral, we can use the power rule for integration. Integrating x^3 gives us (1/4)x^4. Evaluating this integral over the interval [0,2]:
Area = (1/4)(2^4) - (1/4)(0^4)
= (1/4)(16)
= 4 square units
Therefore, the area of region R is 4 square units.
b. To find the volume of the solid formed by rotating region R about the y-axis using vertical slices, we need to use the method of cylindrical shells. Each cylindrical shell has a radius equal to the x-coordinate and a height equal to the infinitesimal change in y.
The volume of each cylindrical shell can be calculated using the formula: volume = 2πrh, where r is the radius and h is the height.
In this case, the radius is x, and since we are rotating about the y-axis, the height is dy.
Therefore, the volume of the solid can be obtained by integrating the product of the circumference of each shell (2πx) and the height (dy) over the given y-range.
Volume = ∫[0,8] 2πxy dy
To express x in terms of y, we need to solve the equation y = x^3 for x: x = y^(1/3). Substituting this into the volume integral:
Volume = ∫[0,8] 2π(y^(1/3))y dy
Integrating this expression will give you the volume of the solid.
c. The first moment of area of region R with respect to the y-axis represents the tendency of an area to rotate about an axis. For this region, the first moment of area can be calculated by integrating the product of the area (x^3) and the distance from the y-axis (x) with respect to y.
First moment of area = ∫[0,8] x(x^3) dy
Simplifying the integral:
First moment of area = ∫[0,8] x^4 dy
Integrating this expression will give you the first moment of area.
As for noticing something about the integral, it's worth noting that the first moment of area integral does not depend on the y-values of the function. The result is solely dependent on powers of x. This is because the y-axis is the axis of revolution, and the first moment of area does not change with the y-coordinate.
d. To find the x-coordinate of the centroid of region R, we need to calculate the average value of x over the given y-range. The formula for the x-coordinate of the centroid is given by:
x-coordinate of centroid = (First moment of area) / (Area)
Using the values we have previously calculated for the first moment of area and the area of region R:
x-coordinate of centroid = (First moment of area) / (Area)
= ∫[0,8] x^4 dy / 4
Integrating this expression and dividing by 4 will give you the x-coordinate of the centroid.
e. The theorem of Pappus states that the volume of a solid of revolution equals the area of the region being rotated times the distance the centroid of the region travels. In this problem, we can verify this theorem.
We have already calculated the volume of the solid and the area of region R. Now, we need to calculate the distance the centroid of the region travels.
The distance the centroid of the region travels is equal to the x-coordinate of the centroid we found in part d.
So, to verify the theorem, we need to calculate the product of the area of region R and the distance the centroid of the region travels:
Area of region R * Distance the centroid travels
= 4 * (x-coordinate of centroid)
If the result of this calculation matches the previously calculated volume of the solid, then it confirms the theorem of Pappus.