Did the book misprint or am I doing something wrong?

[kyin01]

this is the question
find the volume of solid under the graph f(x,y) = x+ y above the region y^2 > x and 0 < x < 9

So pretty much I have the following integral set up

0 -> 9 dx
-sqrt(X) -> sqrt (X) dy
integrating over the function x+y

However, when I look at the solutions they did the same thing except for their dy, they took it over
0 -> sqrt (X)

is that a misprint or did I do something wrong? Because when I think about it, it's pretty much a parabola and nothing is restricting it from taking each end of the region

Where as the solutions is only taking half the parabola (from 0 to 9 along X axis which is agreed)

so is the solutions a misprint or did I do something wrong?

 

[HallsofIvy]

Unfortunately, it's hard to tell exactly what was intended since you are clearly not copying exactly what the problem said: "the region y^2> x and 0< x< 9" makes no sense. Perhaps they said "the region bounded by y^2= x, x> 0, x< 9"? Or did they possibly say "in the first quadrant"?

 

[kyin01]

sorry it's

Find the volume of the region under the graph f(x,y)= x+y and above the region y^2 < x , 0 < x < 9

that is all they say

 

[Roberto Bramb]

If 0<x<9, then 0<y<3,
so integrate in these limits (x+y) dx dy = 162

 

[D H]

If 0<x<9, then 0<y<3,
so integrate in these limits (x+y) dx dy = 162

No. You are ignoring that the y limits are a function of x.

However, when I look at the solutions they did the same thing except for their dy, they took it over 0 -> sqrt (X)

That suggests a misprint or a misreading. Are you sure the problem doesn't say "above the region y^2 < x"? Even then, the y-integral should be from $-\surd x$ to $\surd x$.