Final answer:
To evaluate the iterated integral ∫[3 to 5] ∫[0 to π/2] x * sin(y) dy dx, we can evaluate the inner integral with respect to y first, and then the outer integral with respect to x. The value of the iterated integral is 8.
Step-by-step explanation:
To evaluate the iterated integral ∫[3 to 5] ∫[0 to π/2] x * sin(y) dy dx, we can use the properties of iterated integrals. First, we integrate the inner integral with respect to y. ∫[0 to π/2] x * sin(y) dy = -x * cos(y) evaluated from y = 0 to y = π/2 = -x * (cos(π/2) - cos(0)) = -x * (0 - 1) = x
Now, we integrate the outer integral with respect to x. ∫[3 to 5] x dx = (1/2) * x^2 evaluated from x = 3 to x = 5 = (1/2) * (5^2 - 3^2) = (1/2) * (25 - 9) = 8
Therefore, the value of the iterated integral is 8.