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Section 1
Introduction: Areas and Volumes
Evaluate the iterated integrals.$\int_{0}^{2} \int_{1}^{3}\left(x^{2}+y\right) d y d x$
Evaluate the iterated integrals.$\int_{0}^{\pi} \int_{1}^{2} y \sin x d y d x$
Evaluate the iterated integrals.$\int_{-2}^{4} \int_{0}^{1} x e^{y} d y d x$
Evaluate the iterated integrals.$\int_{0}^{\pi / 2} \int_{0}^{1} e^{x} \cos y d x d y$
Evaluate the iterated integrals.$\int_{1}^{2} \int_{0}^{1}\left(e^{x+y}+x^{2}+\ln y\right) d x d y$
Evaluate the iterated integrals.$\int_{1}^{9} \int_{1}^{e} \frac{\ln \sqrt{x}}{x y} d x d y$
Find the volume of the region that lies under the graph of the paraboloid $z=x^{2}+y^{2}+2$ and over the rectangle $R=\{(x, y) \mid-1 \leq x \leq 2,0 \leq y \leq 2\}$ in two ways:(a) by using Cavalieri's principle to write the volume as an iterated integral that results from slicing the region by parallel planes of the form $x=$ constant;(b) by using Cavalieri's principle to write the volume as an iterated integral that results from slicing the region by parallel planes of the form $y=$ constant.
Find the volume of the region bounded on top by the plane $z=x+3 y+1,$ on the bottom by the $x y$ -plane, and on the sides by the planes $x=0, x=3, y=1$, $y=2$.
Find the volume of the region bounded by the graph of $f(x, y)=2 x^{2}+y^{4} \sin \pi x,$ the $x y$ -plane, and the planes $x=0, x=1, y=-1, y=2$.
Calculate the given iterated integrals and indicate of what regions in $\mathbf{R}^{3}$ they may be considered to represent the volumes.$\int_{0}^{2} \int_{1}^{3} 2 d x d y$
Calculate the given iterated integrals and indicate of what regions in $\mathbf{R}^{3}$ they may be considered to represent the volumes.$\int_{1}^{3} \int_{-2}^{2}\left(16-x^{2}-y^{2}\right) d y d x$
Calculate the given iterated integrals and indicate of what regions in $\mathbf{R}^{3}$ they may be considered to represent the volumes.$\int_{-\pi / 2}^{\pi / 2} \int_{0}^{\pi} \sin x \cos y d x d y$
Calculate the given iterated integrals and indicate of what regions in $\mathbf{R}^{3}$ they may be considered to represent the volumes.$\int_{0}^{5} \int_{-2}^{2}\left(4-x^{2}\right) d x d y$
Calculate the given iterated integrals and indicate of what regions in $\mathbf{R}^{3}$ they may be considered to represent the volumes.$\int_{-2}^{3} \int_{0}^{1}|x| \sin \pi y d y d x$
Calculate the given iterated integrals and indicate of what regions in $\mathbf{R}^{3}$ they may be considered to represent the volumes.$\int_{-5}^{5} \int_{-1}^{2}(5-|y|) d x d y$
Suppose that $f$ is a nonnegative-valued, continuous function defined on $R=\{(x, y) \mid a \leq x \leq b, c \leq$ $y \leq d\}$. If $f(x, y) \leq M$ for some positive number $M$ explain why the volume $V$ under the graph of $f$ over $R$ is at most $M(b-a)(d-c)$ .