As a hardworking student, plagued by too much homework, you spend all night doing math homework. By 6am, you imagine yourself to be a region bounded by
y=5x2
x=0
x=3
y=0
As you grow more and more tired, the world begins to spin around you. However, according to Newton, there is no difference between the world spinning around you, and you spinning around the world. Unfortunately, you are so tired that you think the world is the x-axis. What is the volume of the solid you (the region) create by spinning about the x-axis?
To find the volume of the solid when the region bounded by the curve y=5x^2, x=0, x=3, and y=0 is rotated about the x-axis, you can use the method of cylindrical shells.
First, visualize the region bounded by the curve y=5x^2, x=0, x=3, and y=0. This is a parabolic shape with the x-axis as the base.
To find the volume, we break the region into small vertical strips (shells) perpendicular to the x-axis. Each shell has a small height and thickness (dx) and can be thought of as a cylindrical shape.
The height of each cylindrical shell is the difference between the upper curve (y=5x^2) and the lower curve (y=0), so it is given by (5x^2 - 0) = 5x^2.
The circumference (or length) of each shell is given by 2π times the radius, which is the x-coordinate. So, the circumference is 2πx.
The volume of each shell is given by the product of its height, its circumference, and its thickness (dx), which is (5x^2)(2πx)(dx) = 10πx^3 dx.
To find the total volume of the solid, we integrate the volume of each shell from x=0 to x=3:
V = ∫(0 to 3) 10πx^3 dx
Evaluating this integral will give us the volume of the solid you create by spinning about the x-axis.
Do remember to evaluate definite integrals as a final step!
To find the volume of the solid formed by spinning the region bounded by y = 5x^2, x = 0, x = 3, and y = 0 about the x-axis, we can use the method of cylindrical shells.
Step 1: Rewrite the equation in terms of y to determine the limits of integration.
From y = 5x^2, we can solve for x:
x = √(y/5)
Since the region is bounded by x = 0 and x = 3, its limits of integration with respect to y are:
0 ≤ y ≤ 5(3^2)
0 ≤ y ≤ 45
Step 2: Determine the height or length of each cylindrical shell.
The height or length of each shell is given by the function y = 5x^2.
Step 3: Determine the radius of each cylindrical shell.
The radius of each shell is given by the distance from the x-axis to the curve y = 5x^2, which is x.
Step 4: Determine the thickness or width of each cylindrical shell.
The thickness or width is given by dy since we are integrating with respect to y.
Step 5: Set up the integral to calculate the volume using the formula for the volume of cylindrical shells.
The volume of each cylindrical shell is given by 2πrhdy, where r is the radius, h is the height, and dy is the thickness.
The equation to calculate the volume is:
V = ∫[0,45] 2πx(5x^2)dy
Step 6: Simplify the integral and evaluate.
V = 2π ∫[0,45] 5x^3 dy
We can substitute x = √(y/5) into the integral:
V = 2π ∫[0,45] 5(√(y/5))^3 dy
= 10π ∫[0,45] y^(3/2) dy
Integrating y^(3/2):
V = 10π (2/5) y^(5/2) |[0,45)
= 10π [(2/5) (45)^(5/2) - (2/5)(0)^(5/2)]
= 10π [(2/5) (45)^2.5 - 0]
Calculating the final result:
V ≈ 10π (2/5)(299.83)
V ≈ 5976π
Therefore, the volume of the solid formed by spinning the region around the x-axis is approximately 5976π cubic units.