91Ó°ÊÓ

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 \(y=x^{2}\) and \(y=x+2\)

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{-1}^{1} \int_{-\sqrt{1-y^{2}}}^{\sqrt{1-y^{2}}} \ln \left(x^{2}+y^{2}+1\right) d x d y$$

Find the average height of the paraboloid \(z=x^{2}+y^{2}\) over the square \(0 \leq x \leq 2,0 \leq y \leq 2\).

Find the average value of \(f(x, y)=1 /(x y)\) over the square \(\ln 2 \leq x \leq 2 \ln 2, \ln 2 \leq y \leq 2 \ln 2\).

Sketch the region of integration and evaluate the integral. $$\int_{1}^{2} \int_{y}^{y^{2}} d x d y$$

Centroid of a solid semiellipsoid Assuming the result that the centroid of a solid hemisphere lies on the axis of symmetry threeeighths of the way from the base toward the top, show, by transforming the appropriate integrals, that the center of mass of a solid semiellipsoid \(\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)+\left(z^{2} / c^{2}\right) \leq 1, z \geq 0\) lies on the \(z\) -axis three-eighths of the way from the base toward the top. (You can do this without evaluating any of the integrals.)

The tetrahedron in the first octant bounded by the coordinate planes and the planes passing through \((1,0,0),(0,2,0),\) and (0,0,3) (GRAPH CAN'T COPY).

The region in the first octant bounded by the coordinate planes, the plane \(y=1-x,\) and the surface \(z=\cos (\pi x / 2), 0 \leq x \leq 1\). (GRAPH CAN'T COPY).

Find the area of the region cut from the first quadrant by the cardioid \(r=1+\sin \theta.\)

Use Fubini's Theorem to evaluate $$\int_{0}^{2} \int_{0}^{1} \frac{x}{1+x y} d x d y$$