91Ó°ÊÓ

Find the area of the part of the paraboloid \(z=9-x^{2}-y^{2}\) that lies above the plane \(z=5\)

Use double integration to find the area of the region in the xy-plane bounded by the given curves. $$y=x^{2}, y=2 x+3$$

Find the indicated area by double integration in polar coordinates. The area bounded by the cardioid \(r=1+\cos \theta\) (Fig. 14.4.16)

Solve for \(x\) and \(y\) in terms of \(u\) and \(v\). Then compute the Jacobian \(\partial(x, y) / \partial(u, v)\) $$u=x y, \quad v=y / x$$

Use double integration to find the area of the region in the xy-plane bounded by the given curves. $$y=2 x+3, y=6 x-x^{2} \quad$$

Find the centroid of the plane region bounded by the given curves. Assume that the density is \(\delta \equiv 1\) for each region. $$x=0, y=0, x+y=3$$

Solve for \(x\) and \(y\) in terms of \(u\) and \(v\). Then compute the Jacobian \(\partial(x, y) / \partial(u, v)\) $$u=2\left(x^{2}+y^{2}\right), \quad v=2\left(x^{2}-y^{2}\right)$$

Calculate the Riemann sum for $$ \iint_{R} f(x, y) d A $$ using the given partition and selection of points \(\left(x_{t}^{*}, y_{t}^{*}\right)\) for the rectangle \(R\). \(f(x, y)=x y ; R=|0.2| \times[0.2]\) : the partition \(P\) consists of four unit squares: each \(\left(x_{t}^{*}, y_{t}^{*}\right)\) is the center point of the \(i\) th rectangle \(K_{i}\).

Find the area of the part of the surface \(2 z=x^{2}\) that lies directly above the triangle in the \(x y\) -plane with vertices at \((0,0),(1,0)\), and \((1,1)\)

Find the indicated area by double integration in polar coordinates. The area bounded by one loop of \(r=2 \cos 2 \theta\) (Fig. 14.4.17)