Evaluate the triple integral ∫∫∫_E (x+y)dV where E is bounded by the parabolic cylinder y=5x^2 and the planes z=9x, y=20x and z=0.
FSF
To evaluate the triple integral ∫∫∫_E (x+y)dV, we need to find the limits of integration for each variable (x, y, and z) and set up the integral using these limits.
First, let's analyze the given region E which is bounded by the parabolic cylinder y = 5x^2, and the planes z = 9x, y = 20x, and z = 0.
Since the parabolic cylinder and the planes intersect at certain points, we need to find the limits of integration by determining the boundaries of E.
To find the limits of integration for x, we need to determine the x-values where the parabolic cylinder and the planes intersect.
1. Intersection of parabolic cylinder and z = 9x:
Setting 5x^2 = 9x, we get 5x^2 - 9x = 0.
Factoring out x, we have x(5x - 9) = 0.
This gives us two possible solutions: x = 0 and x = 9/5.
2. Intersection of parabolic cylinder and y = 20x:
Setting 5x^2 = 20x, we get 5x^2 - 20x = 0.
Factoring out 5x, we have 5x(x - 4) = 0.
This gives us two possible solutions: x = 0 and x = 4.
Since x = 0 is a common intersection for both cases, we will only consider x = 0 and x = 9/5 as the limits of integration for x.
Next, let's consider the limits of integration for y.
The parabolic cylinder y = 5x^2 bounds the region E, so the limits of y should be defined by this equation.
Since we are given y = 5x^2, we can write this as y = 5(x^2) and take the square root of both sides:
√y = √(5x^2)
√y = x√5
Now, we need to solve for x in terms of y:
x = √y/√5
Using the boundaries for x, we can determine the corresponding boundaries for y:
When x = 0, y = 5(0) = 0.
When x = 9/5, y = 5((9/5)^2) = 45/5 = 9.
Thus, the limits of integration for y are from y = 0 to y = 9.
Finally, let's consider the limits of integration for z.
The planes z = 9x and z = 0 bound the region E, so the limits of z should be defined by these equations.
From z = 9x, we can express x in terms of z:
x = z/9
Using the boundaries for x, we can determine the corresponding boundaries for z:
When x = 0, z = 0.
When x = 9/5, z = (9/5)/9 = 1/5.
Thus, the limits of integration for z are from z = 0 to z = 1/5.
Now that we have determined the limits of integration for x, y, and z, we can set up the triple integral as follows:
∫∫∫_E (x+y)dV = ∫∫∫_E (x+y) dx dy dz
∫_(0 to 9/5) ∫_(0 to 9) ∫_(0 to 1/5) (x+y) dz dy dx
Evaluate this triple integral to find the numerical value.
To evaluate the triple integral ∫∫∫_E (x+y)dV, where E is bounded by the parabolic cylinder y=5x^2 and the planes z=9x, y=20x, and z=0, we can follow these steps:
Step 1: Visualize the region E
Start by visualizing the region E in 3D space. The parabolic cylinder y=5x^2 is a curved surface, and the planes z=9x, y=20x, and z=0 are flat surfaces. Together, they form a solid region E.
Step 2: Determine the limits of integration
To set up the integral, we need to determine the limits of integration for each variable (x, y, z) over the region E.
Since y=5x^2 bounds the region E in the y-axis direction, we can express the limits of y as a function of x: 0 ≤ y ≤ 5x^2.
The planes z=9x, y=20x, and z=0 define the limits of z and x. We need to find the intersection points between these planes and the parabolic cylinder.
The intersection of y=5x^2 and z=0: Substitute z=0 into y=5x^2: 0=5x^2 → x=0.
The intersection of y=5x^2 and z=9x: Substitute z=9x into y=5x^2: 5x^2=9x → x=0, x=9/5.
The intersection of y=5x^2 and y=20x: 5x^2=20x → x=0, x=4.
Therefore, the limits of x are 0 ≤ x ≤ 4, and the limits of z are 0 ≤ z ≤ 9x.
Step 3: Set up and evaluate the triple integral
Now that we have determined the limits of integration, we can set up the triple integral:
∫∫∫_E (x+y)dV = ∫∫∫_E (x+y) dz dy dx
The order of integration can be chosen in any order, but we will integrate in the order dz dy dx.
The integral becomes:
∫(0 to 4) ∫(0 to 5x^2) ∫(0 to 9x) (x+y) dz dy dx
Evaluating the innermost integral, we get:
∫(0 to 4) ∫(0 to 5x^2) (9xz + yz) dy dx
Next, evaluating the middle integral:
∫(0 to 4) (9xz + (5x^2)(z)) dy dx
Finally, evaluating the outer integral:
∫(0 to 4) [9xz + (5x^2)(z)] dx
This integral can be evaluated using standard integration techniques.
Note: The actual numerical evaluation of this triple integral requires the specific values of x, y, and z equations and limits.