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Evaluate ∫⁶⁷₂₆ ∫⁴⁹₀ ∫⁴⁹ᵧ xcos z/z dzdydx.

User Miszmaniac
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2 Answers

4 votes

Answer:

Explanation:

First, we integrate with respect to z. The integral becomes:

∫⁶⁷₂₆ ∫⁴⁹₀ [xsin(z)]₍ᴢ=⁴₉ᵧ₎₌ₓsin(ᵧ)/ᵧ dydx

Next, we integrate with respect to y. The integral becomes:

∫⁶⁷₂₆ [(xsin(⁴₉ᵧ))-(xsin(⁶₇ᵧ))] / ᵧ dx

Finally, we integrate with respect to x. The integral becomes:

[-cos(⁴₉₀) + cos(⁶₇₂₆)] / ᵧ

User Shashi Shankar
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0 votes

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.

User Horkavlna
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