91Ó°ÊÓ

\(19-27\) Use polar coordinates to find the volume of the given solid. Under the cone \(z=\sqrt{x^{2}+y^{2}}\) and above the disk \(x^{2}+y^{2} \leqslant 4\)

Evaluate the integral by making an appropriate change of variables. \(\iint_{R} \frac{x-2 y}{3 x-y} d A,\) where \(R\) is the parallelogram enclosed by the lines \(x-2 y=0, x-2 y=4,3 x-y=1,\) and \(3 x-y=8\)

Consider a square fan blade with sides of length 2 and the lower left corner placed at the origin. If the density of the blade is \(\rho(x, y)=1+0.1 x,\) is it more difficult to rotate the blade about the \(x\) -axis or the \(y\) -axis?

Use a triple integral to find the volume of the given solid. The solid bounded by the cylinder \(y=x^{2}\) and the planes \(z=0, z=4,\) and \(y=9\)

\(15-22\) Calculate the double integral. $$\iint_{R} \frac{x}{1+x y} d A, \quad R=[0,1] \times[0,1]$$

Evaluate the integral by making an appropriate change of variables. \(\iint_{R}(x+y) e^{x^{2}-y^{2}} d A,\) where \(R\) is the rectangle enclosed by the lines \(x-y=0, x-y=2, x+y=0,\) and \(x+y=3\)

\(19-27\) Use polar coordinates to find the volume of the given solid. Below the paraboloid \(z=18-2 x^{2}-2 y^{2}\) and above the \(x y\) -plane

Use cylindrical coordinates. Evaluate \(\iiint_{E} X d V,\) where \(E\) is enclosed by the planes \(z=0\) and \(z=x+y+5\) and by the cylinders \(x^{2}+y^{2}=4\) and \(x^{2}+y^{2}=9\)

Find the volume of the given solid. Under the surface \(z=2 x+y^{2}\) and above the region bounded by \(x=y^{2}\) and \(x=y^{3}\)

Find the volume of the given solid. Under the surface \(z=x y\) and above the triangle with vertices (1.1), \((4,1),\) and \((1,2)\)