10.7: Change of Variables in Multiple Integrals
- Page ID
- 144356
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- Find the Jacobian of the transformation \(x = r\cos \theta\), \(y = r\sin \theta\).
- Find the Jacobian of the transformation \(x = \rho\sin\phi\cos\theta\), \(y = \rho\sin\phi\sin\theta\), \(z = \rho\cos\phi\).
- Let \(R\) be the region in the first quadrant bounded by \(x = 1\), and \(y = x\). Evaluate \(\displaystyle \iint_R (x + y)\ dA\) using the transformation \(u = x+y\), \(v = x-y\).
- Let \(R\) be the triangular region with vertices \((0, 0)\), \((2, 0)\), and \((1, 1)\). Evaluate \(\displaystyle \iint_R \sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)\ dx\,dy\) by making the change of variables \(u = \frac{1}{2}(x+y)\), \(v = \frac{1}{2}(x-y)\).
- Let \(R\) be the parallelogram with vertices \((0, 0)\), \((3, 3)\), \((7, 3)\), and \((4, 0)\). Evaluate \(\displaystyle \iint_R y(x - y)\ dA\) using the transformation \(u = x-y\), \(v = y\).
- Let \(R\) be the region in the first quadrant enclosed by \(y = x\), \(y = 3x\), \(xy = 1\), and \(xy = 4\). Evaluate \(\displaystyle \iint_R xy^3\ dx\,dy\) by making the change of variables \(u = xy\), \(v = y/x\).
- Let \(R\) be the region bounded by the ellipse \(x^2 + 25y^2 = 1\). Evaluate \(\displaystyle \iint_R \sqrt{x^2 + 25y^2}\ dA\) using the transformation \(u = x\), \(v = 5y\).
- Let \(E\) be the solid enclosed by the ellipsoid \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2} = 1\). Evaluate \(\displaystyle \iiint_E \ dV\) using the transformation \(u = \dfrac{x}{a}\), \(v = \dfrac{y}{b}\), \(w = \dfrac{z}{c}\).