- Compute $\int_0^3\int_x^3\cos(y^2)dydx$. Hint: Draw the region, swap the order of the bounds, then integrate.
- Consider the region $R$ in the first quadrant that is above the line $y=x$ and below the circle $x^2+y^2=25$. Set up integral formulas using iterated double integrals to find the centroid $(\bar x,\bar y)$ of $R$.
- Set up the integrals using rectangular coordinates.
- Set up the integrals using polar coordinates. [Note: Recall that $dA=rdrd\theta$ in polar coordinates.]
- Consider the iterated triple integral $$V=\iiint_D dV=\int_{-3}^{3}\int_{0}^{\sqrt{9-y^2}}\int_{0}^{9-x^2-y^2}dzdxdy.$$ This integral gives the volume of the solid $D$ in space described by the bounds of the integral.
- Draw the region described by the bounds of this integral.
- Set up the integral using the order of integration $dydxdz$.
- Set up the integral using the order of integration $dxdydz$.
- Consider the iterated triple integral $$V=\iiint_D dV=\int_{-1}^{1}\int_{0}^{1-x^2}\int_{0}^{y}dzdydx.$$
- Draw the region described by the bounds of this integral.
- Set up the integral using the order of integration $dydzdx$.
- Set up the integral using the order of integration $dxdydz$.
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