Consider the solid obtained by rotating the region bounded by the given curves about the x-axis.
y = 9 - 9x^2 , y = 0
Find the volume V of this solid.
Sketch the region, the solid, and a typical disk or washer.
Any help or tips would be greatly appreciated; not sure how to work it :/
The curve intersects the x-axis at (1,0) and (-1,0).
Volume = π[integral] y^2 dx from -1 to 1
or by symmetry
= 2π[integral] (9-9x^2)^2 dx from 0 to 1
= 2π[integral] (81 - 162x^2 + 81x^4) dx from 0 to 1
= 2π(81x - 54x^3 + (81/5)x^5) from 0 to 1
= 2π(81 - 54 + 81/5 - 0)
= 432π/5
Well, well, well, we have ourselves a rotational solid problem, don't we? Time to put on my thinking clown nose!
First things first, let's take a look at that region we're dealing with. The curves y = 9 - 9x^2 and y = 0. Hmm, let's sketch it out.
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____/______/
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Okay, so that's a terrible sketch, but you get the idea, right? We have a curve and a flat line, and we want to rotate this region about the x-axis!
To find the volume of this solid, we can break it down into infinitesimally thin disks (or washers, but I'll stick with disks because they're easier to visualize). Each disk will have a radius and a height, and we can sum up the volume of all those disks to find the total volume.
Now, to calculate the volume of one of those disks, we can use the formula for the volume of a cylinder: V = πr^2h.
The radius of each disk will be the value of y (which is the distance from the curve to the x-axis) at any given x, and the height of each disk will be the thickness of those disks, which we will call Δx.
To find the radius at any given x, we just substitute the value of x into the equation y = 9 - 9x^2. So, the radius, or the distance from the curve to the x-axis, is 9 - 9x^2.
Now, we need to find the limits of integration, which are the x-values where the curves intersect. In this case, y = 0 means the curve crosses the x-axis, so we need to solve 9 - 9x^2 = 0 to find those x-values. Solving that bad boy gives us x = ±1.
Great! We have all the ingredients we need. Now, we just integrate the volume formula from x = -1 to x = 1 to get the total volume:
V = ∫[from -1 to 1] π(9 - 9x^2)^2 dx
Now, solving this integral may require a bit of clown magic (also known as calculus), but I'm confident you can handle it! Once you integrate and evaluate the integral, you'll have your answer.
And voila! You've found the volume of that solid by using the power of rotational disks. Now go impress your friends with your newfound knowledge of solid geometry and make them laugh while you're at it!
To find the volume of the solid obtained by rotating the region bounded by the curves y = 9 - 9x^2 and y = 0 about the x-axis, we can use the method of cylindrical shells.
Step 1: Sketching the region
To start, let's sketch the region bounded by the curves. The curve y = 9 - 9x^2 is a parabola that opens downward and intersects the x-axis at x = 1 and x = -1. The curve y = 0 represents the x-axis. So, the region of interest is the area between x = -1 and x = 1 below the x-axis.
Step 2: Sketching the solid
When we rotate this region about the x-axis, it forms a solid with a hole in the center. The solid looks like a donut shape or a torus.
Step 3: Considering a typical disk or washer
To find the volume, let's consider a typical disk (or washer) located at a distance x from the y-axis. We will approximate the volume of this disk and sum up these volumes to get the total volume of the solid.
- The radius of the disk is x (the distance from the y-axis to the point on the curve).
- The height (or thickness) of the disk is the difference between the y-values of the two curves at that x-value. In this case, it will be y = 9 - 9x^2 minus y = 0, which simplifies to y = 9 - 9x^2.
Step 4: Calculating the volume
The volume of each disk can be calculated using the formula for the volume of a cylinder: V = π * (radius)^2 * height.
So, the volume of each disk is V_disk = π * x^2 * (9 - 9x^2).
Step 5: Integrating to find the total volume
To find the total volume, we need to integrate the expression for V_disk from the lower limit x = -1 to the upper limit x = 1.
V = ∫[from -1 to 1] π * x^2 * (9 - 9x^2) dx.
Solving this integral will give us the volume of the solid.
Note: If you find it difficult to solve the integral, you can use a symbolab integral calculator or any other online integral calculator to evaluate the integral and find the value of V.
To find the volume of the solid obtained by rotating the region bounded by the curves about the x-axis, you can use the method of cylindrical shells.
First, let's start by sketching the region bounded by the curves. The region lies between the x-axis and the curve y = 9 - 9x^2. This region is symmetric around the y-axis, and it extends from x = -1 to x = 1. The region is a curve that opens downwards and has its maximum point at the origin (0, 9). The curve intersects the x-axis at x = -1 and x = 1.
Next, let's visualize the solid when it is rotated about the x-axis. The solid will be a three-dimensional shape created by the rotation of the region around the x-axis. It will have a cylindrical shape, and each "slice" of the solid will be a disk or a washer.
To find the volume of the solid, we can integrate the volumes of these individual cylindrical shells over the interval [-1, 1]. Let's consider a small strip or shell at a given x-value within this interval.
The radius of each cylindrical shell would be the distance from the x-axis to the curve at that particular x-value. In this case, the radius would be given by y = 9 - 9x^2.
The height of each cylindrical shell would be the infinitesimally small change in x, denoted as dx. This represents the thickness or width of each shell.
The volume of each cylindrical shell can be calculated using the formula for the volume of a cylinder:
V_cylinder = 2πrh
Where r is the radius of the shell, and h is the height or thickness of the shell.
In our case, the radius r would be equal to y = 9 - 9x^2 and the height h would be equal to dx.
Thus, the volume of each cylindrical shell can be expressed as:
dV = 2π(9 - 9x^2)dx
To find the total volume V, we integrate the expression for the volume of each cylindrical shell over the interval [-1, 1]:
V = ∫[-1,1] 2π(9 - 9x^2)dx
Evaluating this integral will give you the volume of the solid.
I hope this explanation helps you understand the process of finding the volume of a solid obtained by rotating a region about the x-axis.