91Ó°ÊÓ

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

The integrals and sums of integrals Give the areas of regions in the \(x y\) -plane. Sketch each region, label each bounding curve with its equation, and give the coordinates of the points where the curves intersect. Then find the area of the region. $$\int_{-1}^{0} \int_{-2 x}^{1-x} d y d x+\int_{0}^{2} \int_{-x / 2}^{1-x} d y d x$$

Set up the iterated integral for evaluating \(\iiint_{D} f(r, \theta, z) d z r d r d \theta\) over the given region \(D\) \(D\) is the solid right cylinder whose base is the region in the \(x y-\) plane that lies inside the cardioid \(r=1+\cos \theta\) and outside the circle \(r=1\) and whose top lies in the plane \(z=4\)

Center of mass, moment of inertia Find the center of mass and the moment of inertia about the \(y\) -axis of a thin plate bounded by the \(x\) -axis, the lines \(x=\pm 1,\) and the parabola \(y=x^{2}\) if \(\delta(x, y)=7 y+1.\)

Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar integral. $$\int_{-1}^{0} \int_{-\sqrt{1-x^{2}}}^{0} \frac{2}{1+\sqrt{x^{2}+y^{2}}} d y d x$$

Center of mass, moment of inertia Find the center of mass and the moment of inertia about the \(x\) -axis of a thin rectangular plate bounded by the lines \(x=0, x=20, y=-1,\) and \(y=1\) if \(\delta(x, y)=1+(x / 20).\)

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. $$\int_{-1}^{1} \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}} \frac{2}{\left(1+x^{2}+y^{2}\right)^{2}} d y d x$$

Evaluate the integrals. $$\int_{0}^{1} \int_{1}^{\sqrt{e}} \int_{1}^{e} s e^{s} \ln r \frac{(\ln t)^{2}}{t} d t d r d s \quad(r s t-\mathrm{space})$$

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