Find the volume of the given solid.
Under the surface
z = 7xy
and above the triangle with vertices
(1, 1), (4, 1), and (1, 2)
The hypotenuse of the base is the line
y = (7-x)/3
So, add up all the small solids with base dx dy and height z
v = ∫[1,4]∫[1,(7-x)/3] 7xy dy dx
To find the volume of the given solid, we need to set up and evaluate a triple integral over the region bounded by the triangle.
Step 1: Determine the limits for the triple integral.
Let's consider the triangle with vertices (1, 1), (4, 1), and (1, 2). From the given vertices, we can see that the x-coordinate varies from 1 to 4, and the y-coordinate varies from 1 to 2.
So, the limits for the x-coordinate are 1 to 4, and the limits for the y-coordinate are 1 to 2.
Step 2: Set up the triple integral.
The volume can be calculated using the triple integral of the function z = 7xy over the given region. So, the volume V is given by:
V = ∫∫∫ R 7xy dV
Where R represents the region bounded by the triangle.
Step 3: Evaluate the triple integral.
To evaluate the triple integral, we need to convert it into an iterated integral.
The integral can be rewritten as:
V = ∫ from 1 to 4 ∫ from 1 to 2 ∫ from 0 to 7xy dz dy dx
Now, we can evaluate the triple integral:
V = ∫ from 1 to 4 ∫ from 1 to 2 7xy * (7xy - 0) dy dx
(We integrate with respect to z first, which gives (7xy - 0))
V = ∫ from 1 to 4 [7xy * (7xy - 0)] dy dx
(Now we integrate with respect to y)
V = ∫ from 1 to 4 (49x^2y^2 - 0) dy dx
(Simplifying the expression inside the brackets)
V = ∫ from 1 to 4 (49x^2y^2) dy dx
V = 49 ∫ from 1 to 4 (x^2 * ∫ from 1 to 2 (y^2) dy) dx
(Factoring out the constant term)
V = 49 ∫ from 1 to 4 (x^2 * [(y^3)/3] from 1 to 2) dx
(Integrating with respect to y)
V = 49 ∫ from 1 to 4 (x^2 * [((2^3)/3) - ((1^3)/3)]) dx
(Substituting the limits of y into the expression)
V = 49 ∫ from 1 to 4 (x^2 * (8/3 - 1/3)) dx
(Simplifying the expression after substituting)
V = 49 ∫ from 1 to 4 (x^2 * (7/3)) dx
(Simplifying further)
V = 49 * (7/3) ∫ from 1 to 4 x^2 dx
(Pulling out the constant term)
V = 49 * (7/3) * [(x^3)/3] from 1 to 4
(Integrating with respect to x)
V = 49 * (7/3) * ((4^3)/3 - (1^3)/3)
(Substituting the limits of x into the expression)
V = 49 * (7/3) * (64/3 - 1/3)
(Simplifying the expression)
V = 49 * (7/3) * (63/3)
(Simplifying further)
V = 49 * (7/3) * 21
(Performing the multiplication)
V = 7 * 7 * 21
(Simplifying further)
V = 1029
Therefore, the volume of the given solid is 1029 cubic units.
To find the volume of the given solid, you need to calculate the triple integral of the function representing the surface z = 7xy over the region bounded by the triangle with vertices (1, 1), (4, 1), and (1, 2).
The triple integral is calculated as follows:
∫∫∫V f(x, y, z) dV
Since the function is explicitly given as z = 7xy, we can rewrite it as f(x, y, z) = z - 7xy.
Now, let's break down the integral into three parts:
x ranges from 1 to 4,
y ranges from 1 to 2,
and z ranges from 0 to 7xy.
The integral becomes:
∫[1 to 4]∫[1 to 2]∫[0 to 7xy] (z - 7xy) dzdydx
You can integrate in any order you prefer, but for simplicity, let's integrate in the following order:
First, integrate with respect to z, treating x and y as constants:
∫[0 to 7xy] (z - 7xy) dz = (1/2)z^2 - 7xyz ∣[0 to 7xy] = (1/2)(49x^2y^2) - 49x^2y^2 = -23x^2y^2
Now, integrate the above result with respect to y, treating x as a constant:
∫[1 to 2] -23x^2y^2 dy = (-23/3)x^2y^3 ∣[1 to 2] = (-23/3)x^2(8 - 1) = (-161/3)x^2
Lastly, integrate the previous result with respect to x:
∫[1 to 4] (-161/3)x^2 dx = (-161/9)x^3 ∣[1 to 4] = (-161/9)(64 - 1) = (-161/9)(63) = -1127
Therefore, the volume of the given solid is -1127 cubic units.
Note: The negative sign in the result indicates that the volume is oriented in the opposite direction of the positive z-axis.