I need help in starting this question, How do I draw the region R and how do I find the limits?
Thanks in advance for any help.
Below is my (very rough!) interpretation of the region R, the red lines represent the two major planes.
I need help in starting this question, How do I draw the region R and how do I find the limits?
Thanks in advance for any help.
Below is my (very rough!) interpretation of the region R, the red lines represent the two major planes.
Below is the Mathematica code to draw the region. So it's that wedge in there right? Can you look at the code and determine what each curve (blue, light purple, and light red) are?
So if the integral is:
$\displaystyle \int\int\int f dzdydx$
can you look at the plot and determine first what the limits on z are, then the limits on y, then finally x?
Code:poly1 = Graphics3D[Polygon[{{4, 8, 0}, {-4, 8, 0}, {-4, 0, 0}, {4, 0, 0}}]]; cp1 = ContourPlot3D[y == x^3, {x, 0, 2}, {y, 0, 8}, {z, 0, 4}, ContourStyle -> {Opacity[0.4], LightPurple}, Mesh -> None] cp2 = ContourPlot3D[z == x, {x, 0, 4}, {y, 0, 8}, {z, 0, 5}, ContourStyle -> {Opacity[0.4], LightGreen}, Mesh -> None] Show[{poly1, cp1, cp2}, PlotRange -> {{0, 4}, {0, 8}, {0, 5}}, Axes -> True, AxesLabel -> {Style["X", 10], Style["Y", 10], Style["Z", 10]}, AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}}, BoxRatios -> {1, 1, 1}]
Alright, look at the plot, z is going from the bottom surface (z=0) to the top surface of the wedge. But that surface is just z=x. Now look at how we'd integrate over y: It's going from that curve in the x-y plane (y=x^3) to the back side you said is y=8. How about x? It's going from 0 to the point where 8=x^3. Can you see how I'd write:
$\displaystyle V=\int_0^2 \int_{x^3}^8 \int_0^x x\,dzdydx$
How about since I did that one to to it this way then:
$\displaystyle \int\int\int x\,dzdxdy$