Ordinary Differential Equations Questions and Answers – Applications of Triple Integral

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This set of Ordinary Differential Equations Multiple Choice Questions & Answers (MCQs) focuses on “Applications of Triple Integral”.

1. Evaluate ∫∫∫ 12y-8x dV in the region behind y=10-2z and bounded by z=2x, z=5 and x=0.
a) 1
b) \(\frac{35}{63}\)
c) \(\frac{3125}{16}\)
d) \(\frac{3125}{6}\)
View Answer

Answer: d
Explanation: We know
From the boundary conditions,
0 < y < 10-2z
0 < x < \(\frac{z}{2}\)
0 < z < 5
Applying these limits on the Triple Integral as follows
\(\int\int\int_{0}^{10-2z} 12y-8x dy dx dz\)
\(=\int\int_{0}^{\frac{\pi}{2}}6 (10-2z)^2-8x(10-2z) dx dz\)
\(=∫_{0}^{5} 14z^3-130z^2+300z dz\)
\(=\frac{3125}{6}\)
Thus the answer is \(=\frac{3125}{6}\).
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2. Assume a planet having a radius R and its density is expressed as = \(\frac{R+r}{2r}D’\).
a) \(\frac{5\pi D’R^3}{2}\)
b) \(\frac{4\pi D’R^3}{3}\)
c) \(\frac{5\pi D’R^3}{3}\)
d) \(\frac{5\pi D’R^3}{12}\)
View Answer

Answer: c
Explanation: Consider the case of r=R
Then,
D=D’
Where D’ is the surface density of the planet
As D → ∞, r → 0
For finding the mass of the planet, we use the triple integration formula
M=∫∫∫ dV
Converting into spherical co-ordinates, we get
M=\(\int\int\int D’r^2 sin\theta \frac{R+r}{2r} dr d\theta d\theta\)
Applying the limits
0 to π
0 to 2π
0 to R
Solving the Triple Integral we get,
M=\(\frac{5\pi D’R^3}{3}\)
Thus, the mass of the planet is \(\frac{5\pi D’R^3}{3}\).

Sanfoundry Global Education & Learning Series – Ordinary Differential Equations.

To practice all areas of Ordinary Differential Equations, here is complete set of 1000+ Multiple Choice Questions and Answers.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn