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Section Solutions

Basics

  1. \(\frac{1}{4}x^4y - \frac{3}{2}x^2\) and \(4x^3-6x\)

  2. \(e^{xy}(x-\frac{1}{y})\) and \(e^x(1-\frac{1}{x})+e^{-x}(1+\frac{1}{x})\)

  3. \(9/2\)

  4. \(25/4\)

Regions

  1. it's a cone

  2. it's the intersection of two circles

  3. \(r \leq 3\)

  4. \(2 \le r \le 5\) and \(\theta \in [-\pi/2,\pi/2]\)

  5. \(\sqrt{2}-1\)

Integration

  1. \(-705\text{,}\) Fubini's Theorem

  2. \(\frac{1}{2} (e-1)\)

  3. \(-\frac{1}{4}(17-5e^6)\)

  4. \(\approx 11.62\)

  5. \(\approx .346\)

  6. \(6\)

  7. \(\frac{2}{3}\)

  8. \(\frac{24}{5}\)

  9. \(\frac{1}{6}\sin^3(2)\)

  10. \(\dsp \int_0^{81} \int_{\frac{x}{9}}^{\sqrt{x}} e^{xy} dA\)

  11. \(4.5\)

  12. \(\frac{2}{3} + 2\ln(2)\)

  13. \(6\pi\)

Polar and Spherical Coordinate Integration

  1. \(\dsp \frac{\pi}{4}\)

  2. \(\dsp \frac{189\pi}{4}\)

  3. \(\dsp \frac{-64(\sqrt{3}-2)\pi}{3}\)

Coordinate Transformations

  1. \(-2u(\frac{1}{w}-1)-2v+\frac{1}{u}(2v^2-w)+v/u^2\)

  2. \(r\)

  3. \(\rho^2\sin(\phi)\)

  4. \(2\)

  5. \(\dsp \int_{0}^7 \int_{7v-v^2}^{v-v^2/2} \sqrt{u} du dv\)

  6. area of ellipse is \(\pi a b\) in this case \(28\pi\) (just like circle)

  7. volume of ellipsoid is \(4/3 \pi abc\) (just like sphere!)

Triple Integrals, Cylindrical, and Spherical Coordinates

  1. \(2472/5\)

  2. \(\dsp \int_0^1{\int_0^{1-x}{\int_0^{\frac{1-x-z}{2}} \ f(x,y,z) \ {dy\ dz\ dx}}} \\ \int_0^1{\int_0^{1-z}{\int_0^{\frac{1-x-z}{2}} \ f(x,y,z) \ {dy\ dx\ dz}}} \\ \int_0^{1}{\int_0^{\frac{1-z}{2}}{\int_0^{1-z-2y} \ f(x,y,z)\ {dx\ dy\ dz}}}\)

  3. \(\dsp \int_0^3{\int_0^{-2x+6}{\int_0^{\frac{-4x-2x+12}{3}} \ f(x,y,z) \ {dy\ dx\ dz}}} \\ \int_0^4{\int_0^{-\frac{3}{4}y+3}{\int_0^{\frac{-4z-3y+12}{2}} \ f(x,y,z) \ {dx\ dz\ dy}}}\)

  4. \(\dsp \int_0^\frac{1}{3}{\int_0^{\sqrt{\frac{1-9y^2}{2}}}{\int_0^{\frac{\sqrt{1-9y^2-4z^2}}{6}} \ f(x,y,z) \ {dy\ dx\ dz}}}\)

  5. \(8\pi\)

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