91Ó°ÊÓ

Write six different iterated triple integrals for the volume of the rectangular solid in the first octant bounded by the coordinate planes and the planes \(x=1, y=2\) and \(z=3 .\) Evaluate one of the integrals.

Sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The parabolas \(x=y^{2}\) and \(x=2 y-y^{2}\).

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{1} \int_{0}^{3-3 x} \int_{0}^{3-3 x-y} d z d y d x$$

Sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The lines \(y=2 x, y=x / 2,\) and \(y=3-x\)

The area of an ellipse The area \(\pi a b\) of the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\) can be found by integrating the function \(f(x, y)=1\) over the region bounded by the ellipse in the \(x y-\) plane. Evaluating the integral directly requires a trigonometric substitution. An easier way to evaluate the integral is to use the transformation \(x=a u, y=b v\) and evaluate the transformed integral over the disk \(G: u^{2}+v^{2} \leq 1\) in the \(u v\) -plane. Find the area this way.

Write an iterated integral for \(\iint_{R} d A\) over the described region \(R\) using (a) vertical cross-sections, (b) horizontal cross-sections. Bounded by \(y=\sqrt{x}, y=0,\) and \(x=9\)

Find the center of mass of a thin triangular plate bounded by the \(y\) -axis and the lines \(y=x\) and \(y=2-x\) if \(\delta(x, y)=6 x+3 y+3\).

Give the limits of integration for evaluating the integral \(\iiint f(r, \theta, z) d z \, r \, d r d \theta\) as an iterated integral over the region that is bounded below by the plane \(z=0,\) on the side by the cylinder \(r=\cos \theta,\) and on top by the paraboloid \(z=3 r^{2}\).

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{\sqrt{2}}^{2} \int_{\sqrt{4-y^{2}}}^{y} d x d y$$

Evaluate the double integral over the given region \(R\). $$\iint_{R} x y \cos y d A, \quad R: \quad-1 \leq x \leq 1, \quad 0 \leq y \leq \pi$$