## Related questions with answers

Question

In Exercise given below, sketch the region R whose area is given by the double integral. Then change the order of integration and show that both orders yield the same area.

$\int_{-2}^2 \int_0^{4-y^2} d x d y$

Solution

VerifiedStep 1

1 of 3$\begin{align*} A(R) &= \int_{-2}^{2}\int_{0}^{4- y^2}dxdy\\ &= \int_{-2}^{2} [x]_{0}^{4- y^2}dy\\ &=\int_{-2}^{2} \left(4-y^2\right) dy\\ &=\left[4y - \dfrac{y^3}{3} \right]_{-2}^{2}\\ &=\left(8 - \dfrac{8}{3}\right) - \left(-8 - \dfrac{-8}{3}\right)\\ &=\dfrac{32}{3} \end{align*}$

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