Calculate the Volume of a Prism with Base Area Bounded by y = x^4 and y = x

In summary, to find the volume of the prism with a base bounded by the curves y=x^4 and y=x, and a top surface of f(x,y)=x+xy, you need to integrate the height (f(x,y)) over the region between the two curves. This can be done by taking the x range from 0 to 1 and determining the corresponding y range for each x value. Then, integrate the expression with respect to y using the appropriate y limits for each x value, and finally integrate the resulting expression with respect to x using the x limits of 0 and 1. This will give you the volume of the prism, which is approximately 0.2417 units^3.
  • #1
Gwilim
126
0
Find the volume of the prism whose base is the area in the [tex]xy[/tex] plane bounded by the curves [tex]y = x^4[/tex] and [tex]y=x[/tex] and whose top is the surface [tex]f(x,y) = x+xy[/tex]

this is what I have so far

area of base = [tex]\int\ x^4dx - \int\ xdx[/tex]
= [tex]$\frac{x^5}{5} - \frac{x^2}{2} + C[/tex]

however I don't know what is meant by the top. I might guess it would represent the height of the prism and just multiply the expression by the one I have for the area, but that doesn't seem right, and this question is worth 10 marks
 
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  • #2
It's not right. You find the volume by integrating the height over the region between the two curves.
 
  • #3
Hmm looks like my tex got a bit messed up on that first post, but it's not hard to see what I did there.

Dick said:
It's not right. You find the volume by integrating the height over the region between the two curves.

What exactly does this mean? Is the f(x,y) function the height? Also, the region between the two curves is an area, how do I integrate over that?
 
  • #4
You integrate f(x,y)*dx*dy. The substance to the problem is figuring out what the x and y limits are. Draw a sketch of the region between the two curves.
 
  • #5
The enclosed region lies between (0,0) and (1,1) on the xy plane. So using 0 and 1 as limits for both x and y, this question simply becomes the double integral of x+xy? Doing this quickly I get an answer of 3/4, is this correct?
 
  • #6
Putting x and y limits both to 0 and 1 means you are integrating over a rectangle in the xy plane. That's not what you want. Let's take the x range from 0 to 1. Now pick a value of x. Look at your sketch and tell me what the corresponding y range is. It will depend on x.
 
  • #7
I thought that was too simple. Ok, so if for example I take [tex]x_0=0.5[/tex] then [tex]y_0[/tex] is in the range 1/16 to 1/2. What now? Since the shape is a prism, would it work if I took the volume produced by integrating over the rectangle, divided by the area of the rectangle and multiplied by the area of the bounded region? Though even if that were to work, it's probably not the method the examiners are looking for.
 
  • #8
No. Wouldn't work anyway. So you got if x=(1/2) then the y range is (1/2)^4 to (1/2). Good. So at a general value of x the y limits are x^4 to x, right? Put those as your dy limits.
 
  • #9
Omigosh I think it just clicked, thankyou. I first integrate with respect to y using x and x^4 as limits, treating each x as a constant. Putting in the limits gives me an expression entirely in x which I can then integrate with respect to x. Using 0 and 1 still for limits of x this gives me an answer of 29/120. have I got it this time?
 

1. What is a prism?

A prism is a three-dimensional shape with two parallel and congruent polygonal faces, called the bases, connected by rectangular faces. It can have various base shapes, such as triangles, rectangles, or hexagons.

2. How do you calculate the volume of a prism?

The volume of a prism can be calculated by multiplying the area of the base by the height of the prism. The formula for calculating the volume of a prism is V = Bh, where B is the base area and h is the height of the prism.

3. What is the base area of a prism?

The base area of a prism is the area of the polygonal shape that makes up the base of the prism. It is usually given in square units, such as square inches or square centimeters.

4. How do you find the base area of a prism with bounded bases?

To find the base area of a prism with bounded bases, you need to first find the points of intersection between the two bounding functions. These points will form the vertices of the polygonal base. Then, you can use the formula for finding the area of a polygon to calculate the base area.

5. Can you provide an example of calculating the volume of a prism with bounded bases?

Yes, for example, if we have a prism with a base bounded by the functions y = x^2 and y = 2x, and a height of 5 units, we can first find the points of intersection by setting the two functions equal to each other: x^2 = 2x. This gives us two solutions, x = 0 and x = 2. These points form the vertices of the base, which is a rectangle with sides of length 2 units and 4 units. Therefore, the base area is 2 x 4 = 8 square units. Then, using the formula V = Bh, we can calculate the volume as V = 8 x 5 = 40 cubic units.

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