91Ó°ÊÓ

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{3} \int_{0}^{\sqrt{9-x^{2}}} \int_{0}^{\sqrt{9-x^{2}}} d z d y d x$$

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 \(x=9\)

Evaluate the integrals in Exercises \(7-20\). $$\int_{0}^{2} \int_{-\sqrt{4-y^{2}}}^{\sqrt{4-y^{2}}} \int_{0}^{2 x+y} d z d x d y$$

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 \(y=\tan x, x=0,\) and \(y=1\)

The integrals and sums of integrals in Exercises \(13-18\) 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. $$\int_{0}^{3} \int_{-x}^{\sqrt{(2-x)}} d y d x$$

Convert the integral $$\int_{-1}^{1} \int_{0}^{\sqrt{1-y^{2}}} \int_{0}^{x}\left(x^{2}+y^{2}\right) d z d x d y$$ to an equivalent integral in cylindrical coordinates and evaluate the result.

Evaluate the iterated integral. $$\int_{-1}^{2} \int_{1}^{2} x \ln y d y d x$$

Find the center of mass and moment of inertia about the \(x\) -axis of a thin plate bounded by the curves \(x=y^{2}\) and \(x=2 y-y^{2}\) if the density at the point \((x, y)\) is \(\delta(x, y)=y+1\).

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 \(y=e^{-x}, y=1,\) and \(x=\ln 3\)

Find the center of mass and the moment of inertia about the \(y\) -axis of a thin rectangular plate cut from the first quadrant by the lines \(x=6\) and \(y=1\) if \(\delta(x, y)=x+y+1\).