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Sagot :
The value of the iterated integral is equal to 24. In general, when evaluating iterated integrals, it is important to remember to integrate with respect to the outer variable first, and then with respect to the inner variable. This will ensure that the resulting value of the integral is correct.
To evaluate this iterated integral, we need to integrate with respect to the outer variable first, and then with respect to the inner variable. In this case, the outer variable is x and the inner variable is y.
We can write the iterated integral as:
∫₀² ∫₂³ xy² dx dy
To find the value of this integral, we first need to evaluate the inner integral with respect to x. We can do this by taking the integral of xy² with respect to x from 0 to 2, which gives us:
∫₀² xy² dx = [x³y² / 3]₀² = (2³ * 3²) / 3 = 24/3
Now that we have found the value of the inner integral, we can evaluate the outer integral with respect to y. We can do this by taking the integral of 24/3 with respect to y from 2 to 3, which gives us:
∫₂³ (24/3) dy = [24y / 3]₂³ = (24 * 3) / 3 = 24
Therefore, the value of the iterated integral is equal to 24.
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