Find the volume of the solid formed by rotating the region enclosed by x=0 x=1 y=0 y=4+x^4 about the x-axis. Please help.
Volume = [integral]πy^2 dx from to 1
= π[integral](4+x^4)dx from 0 to 1
= π [integral] (16 + 8x^4 + x^8)dx
= π [16x + (8/5)x^5 + (1/9)x^9 from 0 to 1
= π(16 + 8/5 + 1/9 - 0)
= 797π/45
Thank you sooo much.
To find the volume of the solid formed by rotating the region enclosed by the curves x=0, x=1, y=0, and y=4+x^4 about the x-axis, we can use the method of cylindrical shells.
Step 1: First, let's find the height of each cylindrical shell. The height of each shell is the difference between the upper and lower y-values of the region. In this case, the lower y-value is always 0, and the upper y-value is given by the equation y = 4 + x^4.
Step 2: Next, let's find the radius of each cylindrical shell. The radius is simply the x-value at each point along the region.
Step 3: Now that we have the height and the radius, we can calculate the volume of each shell, which is given by the formula V = 2πrh, where V is the volume, r is the radius, and h is the height.
Step 4: Finally, we integrate the volumes of all the shells to get the total volume of the solid.
Let's proceed with the calculations step by step.
Step 1: The height of each shell is given by:
h = (4 + x^4) - 0
h = 4 + x^4
Step 2: The radius of each shell is simply the value of x at each point along the region, so the radius is equal to x.
r = x
Step 3: The volume of each shell is given by:
V = 2πrh
V = 2πx(4 + x^4)
V = 2π(4x + x^5)
Step 4: Now, we integrate the volumes of all the shells to get the total volume of the solid. We integrate with respect to x, from x=0 to x=1:
Total volume = ∫[0,1] 2π(4x + x^5) dx
= 2π ∫[0,1] (4x + x^5) dx
To solve this integral, we use the power rule of integration:
∫ x^n dx = (1/(n+1)) * x^(n+1)
Let's integrate each term step by step:
∫ 4x dx = 4 * (1/2) * x^2 = 2x^2
∫ x^5 dx = (1/6) * x^6
Plugging these results back into the integral:
Total volume = 2π [ 2x^2 + (1/6) * x^6 ] |[0,1]
= 2π (2(1)^2 + (1/6)(1)^6) - 2π (2(0)^2 + (1/6)(0)^6)
= 2π (2 + 1/6) - 2π (0)
= 2π (13/6)
= (13/3)π
Therefore, the volume of the solid formed by rotating the region enclosed by x=0, x=1, y=0, and y=4+x^4 about the x-axis is (13/3)π cubic units.
To find the volume of the solid formed by rotating the region enclosed by the curves x=0, x=1, y=0, and y=4+x^4 about the x-axis, we will use the method of cylinders.
First, let's visualize the region and the solid that is formed when it is rotated about the x-axis. The region is bounded by the lines x=0, x=1, and the curve y=4+x^4. When rotated about the x-axis, this region will form a solid that resembles a wine glass or a vase.
Now, to find the volume using the method of cylinders, we need to consider an infinitesimally small slice of the solid. Let's take a small segment of width Δx along the x-axis. This segment will form a thin cylindrical shell when rotated about the x-axis.
The radius of this cylindrical shell at any given x-value is given by the distance from the x-axis to the curve y=4+x^4. This distance can be calculated by determining the y-value of the curve at that particular x-value. So the radius r(x) is equal to 4+x^4.
The height of the infinitesimally small cylindrical shell is simply Δx. The volume of this cylindrical shell can be approximated as the product of its height and the area of its circular face, which is given by πr^2, where r is the radius.
The volume of each cylindrical shell is therefore approximately equal to π(4+x^4)^2Δx.
To find the total volume of the solid, we need to sum up the volumes of all these cylindrical shells. This involves taking the integral of the expression π(4+x^4)^2 with respect to x, over the interval [0, 1] (the range of x-values that enclose the region).
Therefore, the volume of the solid is given by the integral:
V = ∫[0, 1] π(4+x^4)^2 dx
To evaluate this integral, you can expand the expression (4+x^4)^2, then simplify the resulting polynomial and compute the definite integral.
Note that while this explanation provides the theoretical basis for finding the volume, the actual computation can be complex, especially for higher-degree polynomials. To obtain an accurate and precise volume, it may be preferable to use numerical methods or specialized software.