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Alculate the double integral. r x sec2(y) da, r = (x, y) | 0 ≤ x ≤ 8, 0 ≤ y ≤ 4

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Final answer:

The double integral ∫[xsec2(y)] da over the given range can be calculated by integrating first with respect to x and then with respect to y. The final answer is 0.1185.

Step-by-step explanation:

To calculate the double integral ∫[xsec2(y)] da, we can start by defining the limits of integration. The given range for x is 0 ≤ x ≤ 8 and for y it is 0 ≤ y ≤ 4. Next, we can set up the double integral:

xsec2(y) da

From here, we can integrate with respect to x first, and then with respect to y. Let's assume we have the limits of integration reversed. Applying the limits of integration, we have:

∫[xsec2(y)] dx = [x·tan(y)] |08 = (8·tan(y)) - (0·tan(y)) = 8·tan(y)

Next, we integrate with respect to y:

∫ 8·tan(y) dy = 8·ln|sec(y)|

To find the final result, we evaluate this integral with the given limits of integration:

8·ln|sec(y)| |04

By substituting the limits, we get:

8·ln|sec(4)| - 8·ln|sec(0)|

= 8·ln(1.0149) - 8·ln(1)

= 8·ln(1.0149) - 0

= 8·0.0148

= 0.1185

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