Answer:
To evaluate the given triple integral, we'll start by integrating with respect to z, then y, and finally x. Let's break down each step:
Step 1: Integrate with respect to z
The integral ∫ xcos(z)/z dz can be evaluated using the basic integral rules. The antiderivative of xcos(z)/z with respect to z is x*sin(z). Therefore, integrating the expression with respect to z gives:
∫ xcos(z)/z dz = x*sin(z)
Step 2: Integrate with respect to y
Now, we need to integrate the expression from Step 1 with respect to y. Since there are no y terms in the integrand, the integral of x*sin(z) with respect to y is simply:
∫ x*sin(z) dy = x*sin(z) * y
Step 3: Integrate with respect to x
Finally, we integrate the expression from Step 2 with respect to x. Since the limits of integration are given as ∫₆₇₂₆ for x, we can evaluate the integral as follows:
∫₆₇₂₆ (x*sin(z) * y) dx
Integrating x*sin(z) * y with respect to x gives us:
= (1/2) * x² * sin(z) * y
Now, we can substitute the limits of integration for x, which are 6 and 7:
= (1/2) * [7² * sin(z) * y - 6² * sin(z) * y]
Simplifying further, we get:
= (1/2) * (49 - 36) * sin(z) * y
= (1/2) * 13 * sin(z) * y
Since there are no more variables to integrate with respect to, this is the final answer for the given triple integral:
∫⁶⁷₂₆ ∫⁴⁹₀ ∫⁴⁹ᵧ xcos(z)/z dzdydx = (1/2) * 13 * sin(z) * y
Note: The value of ∫⁴⁹₀ indicates that the limits of integration for y are from 0 to 49, and the value of ∫⁴⁹ᵧ indicates that the limits of integration for z are from 0 to 49. These values are used in the calculation of the integral.