#### Question

Sketch the region R whose area is given by the iterated integral. Then switch the order of integration and show that both orders yield the same area.

$∫_0^2∫_0^x dy dx + ∫_2^4∫_0^4-x dy dx$

#### Solutions

VerifiedSolution A

Solution B

#### Step 1

1 of 3#### Step 1

1 of 13The problem defines two integration regions (since we have two iterated integrals).

We determine the first region of integration from the first integral, which can be written as:

$\int_0^2\left(\int_0^x\,dy\right)\,dx.$

Since under the integral first goes $dy$ and then $dx$, the limits of the internal integral are over the variable $y$.

That is, $y$ takes values between:

$0\leq y\leq x.$

The limits of the external integral are the $x$-axis boundaries. Therefore $x$ can take values between:

$0\leq x\leq 2.$

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