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H. Sketch the region common to the intersecting cylinders r2 + y? < a² and r2 + z2 < a?. Find the volume of this region using a triple integral in Cartesian coordinates. Do the integrals by hand on this one - they should work out easily. (Ans: 16a'/3)

H. Sketch the region common to the intersecting cylinders r2 + y? < a² and r2 + z2 < a?. Find the volume of this region using a triple integral in Cartesian coordinates. Do the integrals by hand on this one - they should work out easily. (Ans: 16a'/3)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### H. Volume of Region Common to Intersecting Cylinders **Problem:** Sketch the region common to the intersecting cylinders \( x^2 + y^2 \leq a^2 \) and \( x^2 + z^2 \leq a^2 \). Find the volume of this region using a triple integral in Cartesian coordinates. Do the integrals by hand on this one—they should work out easily. **Solution:** - **Answer:** \(\frac{16a^3}{3}\) ### I. Average Temperature of Cylindrical Region **Problem:** Set up a triple integral in Cartesian coordinates whose value gives the average temperature of the cylindrical region \( W \) bounded by the planes \( z = 0 \), \( z = 2 \), and the cylinder \( x^2 + y^2 = 1 \). The temperature distribution is \( T = f(x, y, z) = 50e^{x^2 + y^2} \). Evaluate using the Wolfram Alpha widget. **Solution:** - **Answer:** 85.9
Transcribed Image Text:### H. Volume of Region Common to Intersecting Cylinders **Problem:** Sketch the region common to the intersecting cylinders \( x^2 + y^2 \leq a^2 \) and \( x^2 + z^2 \leq a^2 \). Find the volume of this region using a triple integral in Cartesian coordinates. Do the integrals by hand on this one—they should work out easily. **Solution:** - **Answer:** \(\frac{16a^3}{3}\) ### I. Average Temperature of Cylindrical Region **Problem:** Set up a triple integral in Cartesian coordinates whose value gives the average temperature of the cylindrical region \( W \) bounded by the planes \( z = 0 \), \( z = 2 \), and the cylinder \( x^2 + y^2 = 1 \). The temperature distribution is \( T = f(x, y, z) = 50e^{x^2 + y^2} \). Evaluate using the Wolfram Alpha widget. **Solution:** - **Answer:** 85.9
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