Question

∫9 0 ∫3 √y √x³+4dxdy The answer should be in the form ∫ba ∫g²(x) g¹(x) f(x,y) dy dx where a and b are the new limits of integration, g1(x) and g2(x) are the new bounds for y, and f(x,y) represents the integrand.

Please provide the new limits of integration and the updated integral expression in the requested form.

Answer 1

The original double integral is transformed by changing the order of integration. The outer limits become 0 to 3 for x, while the inner limits for y are now x² to 3. The updated integral is ∠30 ∠3x² √ x dy dx.

Step-by-step explanation:

The student's double integral is originally ∠90 ∠3√ y √(x³+4) dx dy, which needs to be rearranged into the form ∠ba ∠g²(x)g¹(x) f(x,y) dy dx, with new integration limits and bounds. To find the new bounds for y in terms of x and change the order of integration, we solve the inner limit for y, getting x=√ y which leads to y=x².

The original bounds of y = 0 to y = 9 correspond to x = 0 to x = 3, thus our new outer integration limits are a = 0 and b = 3. The inner bounds of integration are g¹(x) = x² and g²(x) = 3. Therefore, the updated integral expression is ∠30 ∠3x² √ x dy dx, which meets the desired format.