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Isa Trevino

Answered question

2021-11-09

Changing the order of integration Assuming f is integrable, change the order of integration in the following integral.

\int_{0}^{1} \int_{0}^{\sqrt{1 - y^{2}}} f (x, y) d x d y

Maciej Morrow

Skilled2021-11-10Added 98 answers

Step 1
Concept −
If we have an integral of the form −

I = \int_{a}^{b} \int_{c}^{d} f (x, y) d x d y

Then we can write it as −

I = \int_{a}^{b} \int_{c}^{d} f (x, y) d x d y = \int_{c}^{d} \int_{a}^{b} f (x, y) d y d x

Step 2
Given integral −

I = \int_{0}^{1} \int_{0}^{\sqrt{1 - y^{2}}} f (x, y) d x d y

Changing order −

I = \int_{0}^{\sqrt{1 - y^{2}}} \int_{0}^{1} f (x, y) d y d x

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