Let N = \int \int \int_B xyz^2\ dV , where B is the cuboid bounded by the regions 0 \leq x \leq 1, -1 \leq y \leq 2 and 0 \leq z \leq 3. If N = \frac{a}{b}, where a and b are coprime positive integers. What is the value of a+b?
seems pretty straightforward:
v = ∫[0,1]∫[-1,2]∫[0,3] xyz^2 dz dy dx
= ∫[0,1]∫[-1,2] 1/3 xyz^3[0,3] dy dx
= 9∫[0,1]∫[-1,2] xy dy dx
= 9∫[0,1] 1/2 xy^2 [-1,2] dx
= 27/2 ∫[0,1] x dx
= 27/2 (1/2 x^2)[0,1]
= 27/4
To find the value of N, we need to evaluate the triple integral over the given region B.
The given triple integral is:
N = ∫∫∫B xyz^2 dV
First, let's break down the limits of integration for each variable in the integral:
For x:
0 ≤ x ≤ 1
For y:
-1 ≤ y ≤ 2
For z:
0 ≤ z ≤ 3
Now, let's integrate with respect to x, y, and z.
Starting with x, we integrate the function xyz^2 with respect to x, treating y and z as constants:
∫ (xyz^2) dx = (1/2)xy*z^2
Next, we integrate the result obtained in the previous step with respect to y:
∫ [(1/2)xy*z^2] dy = (1/2)(1/2)xy^2*z^2 = (1/4)xy^2*z^2
Finally, we integrate the result obtained in the previous step with respect to z:
∫ [(1/4)xy^2*z^2] dz = (1/4)x*y^2*(1/3)z^3 = (1/12)xyz^3
Now, we can evaluate the triple integral by plugging in the limits of integration:
N = ∫∫∫B xyz^2 dV
= ∫₀³ ∫₋₁² ∫₀¹ (1/12)xyz^3 dx dy dz
Evaluating the integrals, we get:
N = (1/12) ∫₀³ ∫₋₁² [xy^2z^3]₀¹ dz
= (1/12) ∫₀³ [xy^2z^3]₋₁² dz
= (1/12) [(2xy^2(3^3) - 2xy^2(0^3)) - (2xy^2((-1)^3) - 2xy^2(0^3))]
Simplifying further:
N = (1/12) [(2*27xy^2) - (2*(-1)xy^2)]
= (1/12) (54xy^2 + 2xy^2)
= (1/12) (56xy^2)
Now, we need to find the average value of the function over the region B by dividing N by the volume of B. The volume of B, V, is given by the product of the ranges for x, y, and z:
V = (1 - 0) * (2 - (-1)) * (3 - 0)
= 1 * 3 * 3
= 9
The average value of the function is calculated as:
Average value = N / V = (56xy^2) / 12 * 9 = (7xy^2) / 9
Since we want to find the value of a + b, we need to find a and b in the fraction (7xy^2) / 9.
Hence, a = 7 and b = 9.
Therefore, a + b = 7 + 9 = 16.