The region of integration is bounded by the curves 𝑥²+4𝑦²=4 and 𝑥+2𝑦=2 in the first quadrant.

First, we need to rewrite these equations in a more convenient form:

1. 𝑥²+4𝑦²=4 can be rewritten as 𝑦 = sqrt((4 - 𝑥²)/4) = sqrt(1 - 𝑥²/4), since we are only considering the first quadrant where y = 0.

2. 𝑥+2𝑦=2 can be rewritten as 𝑦 = (2 - 𝑥)/2.

Now, we need to find the intersection points of these two curves. Setting the two equations equal to each other gives:

sqrt(1 - 𝑥²/4) = (2 - 𝑥)/2

Solving this equation gives 𝑥 = 0 and 𝑥 = 2.

So, the region of integration is bounded below by 𝑦 = sqrt(1 - 𝑥²/4) and above by 𝑦 = (2 - 𝑥)/2, for 𝑥 in [0, 2].

Therefore, the correct double integral is:

∫ from 0 to 2 ∫ from sqrt(1 - 𝑥²/4) to (2 - 𝑥)/2 𝑓(𝑥,𝑦) 𝑑𝑦 𝑑𝑥

This does not match either of the provided choices, so the correct answer is "None of these".

Question

The region of integration is bounded by the curves 𝑥²+4𝑦²=4 and 𝑥+2𝑦=2 in the first quadrant.

First, we need to rewrite these equations in a more convenient form:

1. 𝑥²+4𝑦²=4 can be rewritten as 𝑦 = sqrt((4 - 𝑥²)/4) = sqrt(1 - 𝑥²/4), since we are only considering the first quadrant where y >= 0.

2. 𝑥+2𝑦=2 can be rewritten as 𝑦 = (2 - 𝑥)/2.

Now, we need to find the intersection points of these two curves. Setting the two equations equal to each other gives:

sqrt(1 - 𝑥²/4) = (2 - 𝑥)/2

Solving this equation gives 𝑥 = 0 and 𝑥 = 2.

So, the region of integration is bounded below by 𝑦 = sqrt(1 - 𝑥²/4) and above by 𝑦 = (2 - 𝑥)/2, for 𝑥 in [0, 2].

Therefore, the correct double integral is:

∫ from 0 to 2 ∫ from sqrt(1 - 𝑥²/4) to (2 - 𝑥)/2 𝑓(𝑥,𝑦) 𝑑𝑦 𝑑𝑥

This does not match either of the provided choices, so the correct answer is "None of these".

Knowee AI · Accepted Answer