Suggest a substitution/transformation that will simplify the following integrands, and find their Jacobians.
Suggest a substitution/transformation that will simplify the following integrands, and find their Jacobians.
327
Let \(D\) be the unit disk: \(x^2+y^2\leq 1\). Evaluate \[ \intop\!\!\!\intop\nolimits_{D} \exp\, (x^2 + y^2)\, {\it dx}\,{\it dy} \] by making a change of variables to polar coordinates.
Let \(D\) be the region \(0 \le y \le x\) and \(0 \le x \le 1\). Evaluate \[ \intop\!\!\!\intop\nolimits_{D} (x+y)\, {\it dx}\, {\it dy} \] by making the change of variables \(x=u+v\), \(y=u-v\). Check your answer by evaluating the integral directly by using an iterated integral.
Let \(T (u,v) = (x (u,v), y (u,v))\) be the mapping defined by \(T (u,v) = (4u,2u + 3v)\). Let \(D^*\) be the rectangle \([0,1] \times [1,2]\). Find \(D = T (D^*)\) and evaluate
by making a change of variables to evaluate them as integrals over \(D^*\).
Repeat the previous exercise for \(T (u,v) = (u,v(1 + u))\).
Evaluate \[ \intop\!\!\!\intop\nolimits_{D} \frac{{\it dx}\,{\it dy}}{{\sqrt{1+x+2y}}}, \] where \(D = [0,1] \times [0,1]\), by setting \(T (u,v) = (u,v/2)\) and evaluating an integral over \(D^*\), where \(T ( D^*) =D\).
Define \(T(u,v) = (u^2 - v^2, 2uv)\). Let \(D^*\) be the set of \((u,v)\) with \(u^2 + v^2 \le 1, u \ge 0, v \ge 0\). Find \(T ( D^*) =D\). Evaluate \({\intop\!\!\intop}_D {\it dx}\,{\it dy}\).
Let \(T (u,v)\) be as in the previous exercise. By making a change of variables, “formally” evaluate the “improper” integral \[ \intop\!\!\!\intop\nolimits_{D} \frac{{\it dx}\,{\it dy}}{\sqrt{x^2+ y^2}}. \] [NOTE: This integral (and the one in the next exercise) is improper, because the integrand \(1/\sqrt{x^2+y^2}\) is neither continuous nor bounded on the domain of integration. (The theory of improper integrals is discussed in Section 16.4.)]
Calculate \(\displaystyle\int\!\!\!\int_R \frac{1}{x+y} {\it dy}\,{\it dx}\), where \(R\) is the region bounded by \(x=0, y=0, x+y =1, x+y =4\), by using the mapping \(T (u,v) = (u- uv, uv)\).
Evaluate \(\displaystyle\int\!\!\!\int_D (x^2 + y^2)^{3/2} {\it dx}\,{\it dy}\), where \(D\) is the disk \(x^2 + y^2 \le 4\).
Let \(D^*\) be a \(v\)-simple region in the \(uv\) plane bounded by \(v = g (u)\) and \(v = h(u) \leq g(u)\) for \(a \le u \le b\). Let \(T {:}\,\, {\mathbb R}^2 \to {\mathbb R}^2\) be the transformation given by \(x =u\) and \(y = \psi (u,v)\), where \(\psi\) is of class \(C^1\) and \(\partial \psi / \partial v\) is never zero. Assume that \(T ( D^*) = D\) is a \(y\)-simple region; show that if \(f \colon\, D \to {\mathbb R}\) is continuous, then \[ \intop\!\!\!\intop\nolimits_{D} f (x,y)\, {\it dx}\,{\it dy} = \intop\!\!\!\intop\nolimits_{D^*} f (u, \psi (u,v)) \Big| \frac{\partial \psi}{\partial v} \Big|\, du {\,d} v. \]
Use double integrals to find the area inside the curve \(r=1 + \sin \theta\).
Integrate \(z e^{x^2 + y^2}\) over the cylinder \(x^2 + y^2 \le 4, 2 \le z \le 3\).
Let \(D\) be the unit disk. Express \(\displaystyle\int\!\!\!\int_D (1+ x^2 + y^2)^{3/2} {\it dx}\,{\it dy}\) as an integral over \([0,1] \times [0, 2 \pi]\) and evaluate.
Using polar coordinates, find the area bounded by the lemniscate \((x^2 + y^2)^2\) \(=\) \(2a^2\) \((x^2\) \(-\) \(y^2)\).
Redo the fifteenth exercise of Section 15.3 using a change of variables and compare the effort involved in each method.
Calculate \(\displaystyle\int\!\!\!\int_R (x+y)^2 e^{x-y} {\it dx}\, {\it dy}\), where \(R\) is the region bounded by \(x+y =1, x+y=4, x-y=-1\), and \(x-y =1\).
Let \(T \colon\, {\mathbb R}^3 \to {\mathbb R}^3\) be defined by \[ T (u,v,w) = ( u \cos v \cos w , u \sin v \cos w , u \sin w). \]
Integrate \(x^2 + y^2 + z^2\) over the cylinder \(x^2 + y^2 \le 2, -2 \le z \le 3\).
328
Evaluate \(\int_0^\infty e^{-4x^2} {\it dx}.\)
Let \(B\) be the unit ball. Evaluate \[ \intop\!\!\!\intop\!\!\!\intop\nolimits_{B} \frac{{\it dx}\,{\it dy}\,{\it dz}}{\sqrt{2+ x^2 + y^2 + z^2}} \] by making the appropriate change of variables.
Evaluate \(\displaystyle\int\!\!\!\int_A [1 / (x^2 + y^2)^2]\, {\it dx}\,{\it dy}\), where \(A\) is determined by the conditions \(x^2 + y^2 \le 1\) and \(x + y \ge 1\).
Evaluate \(\displaystyle\int\!\!\!\int\!\!\!\int_W {\displaystyle\frac{{\it dx}\,{\it dy}\,{\it dz}}{(x^2+ y^2 + z^2)^{3/2}}}\), where \(W\) is the solid bounded by the two spheres \(x^2 + y^2 + z^2 =a^2\) and \(x^2 + y^2 + z^2 =b^2\), where \(0 < b < a\).
Use spherical coordinates to evaluate: \[ \int_{0}^{3} \int_{0}^{\sqrt{9-x^2}} \int_{0}^{\sqrt{9-x^2-y^2}} \frac{\sqrt{x^2+y^2+z^2}}{1+[x^2+y^2+z^2]^2} \,{\it dz}\, {\it dy}\, {\it dx} \]
Let \(D\) be a triangle in the \((x,y)\) plane with vertices \((0,0),(\frac{1}{2},\frac{1}{2}),(1,0)\). Evaluate: \[ {\intop\!\!\!\intop}_{D} \cos \pi \left( \frac{x-y}{x+y} \right) \, {\it dx}\, {\it dy} \] by making the appropriate change of variables.
Evaluate \(\displaystyle\int\!\!\!\int_D x^2 {\it dx}\,{\it dy}\), where \(D\) is determined by the two conditions \(0 \le x \le y\) and \(x^2 + y^2 \le 1\).
Integrate \(\sqrt{x^2 + y^2 + z^2} \,e^{-(x^2+ y^2+z^2)}\) over the solid bounded by the two spheres \(x^2 + y^2 + z^2 =a^2\) and \(x^2 + y^2 + z^2 =b^2\), where \(0 < b < a\).
Evaluate the following by using cylindrical coordinates.
Evaluate \(\displaystyle\int\!\!\!\int_B (x+y)\, {\it dx}\,{\it dy}\), where \(B\) is the rectangle in the \(xy\) plane with vertices at (0, 1), (1, 0), (3, 4), and (4, 3).
Evaluate \(\displaystyle\int\!\!\!\int_D (x+y)\, {\it dx}\,{\it dy}\), where \(D\) is the square with vertices at \((0, 0), (1, 2), (3, 1)\), and \((2, -1)\).
Let \(E\) be the ellipsoid \((x^2 / a^2) + (y^2/b^2) + ( z^2 / c^2) \le 1\), where \(a, b\), and \(c\) are positive.
Using spherical coordinates, compute the integral of \(f ( \rho, \phi, \theta) = 1 / \rho\) over the region in the first octant of \({\mathbb R}^3\), which is bounded by the cones \(\phi = \pi/4\), \(\phi = \hbox{arctan } 2\) and the sphere \(\rho = \sqrt{6}\).
The mapping \(T (u,v) = (u^2 - v^2, 2uv)\) transforms the rectangle \(1 \le u \le 2, 1 \le v \le 3\) of the \(uv\) plane into a region \(R\) of the \(xy\) plane.
Let \(R\) denote the region inside \(x^2 + y^2 =1\), but outside \(x^2 + y^2 = 2y\) with \(x \ge 0, y \ge 0\).
Let \(D\) be the region bounded by \(x^{3/2}+ y^{3/2} = a^{3/2}\), for \(x \ge 0, y \ge 0,\) and the coordinate axes \(x=0, y=0\). Express \(\displaystyle\int\!\!\!\int_D f (x,y)\, {\it dx}\, {\it dy}\) as an integral over the triangle \(D^*\), which is the set of points \(0 \le u \le a, 0 \le v \le a -u\). (Do not attempt to evaluate.)
Show that \(S( \rho, \theta, \phi) = (\rho \sin \phi\cos\theta\), \(\rho \sin \phi \sin \theta\), \(\rho \cos \phi)\), the spherical change-of-coordinate mapping, is one-to-one except on a set that is a union of finitely many graphs of continuous functions.