**Problem 4:**Let $ f: \mathbb{R}^n \to \mathbb{R} $ be defined by $ f(x_1, x_2, \ldots, x_n) = x_1 x_2 \cdots x_n $ on the cube $[0, 1] \times [0, 1] \times \cdots \times [0, 1]$ (i.e. for $0 \leq x_1 \leq 1, 0 \leq x_2 \leq 1, \ldots, 0 \leq x_n \leq 1$). Evaluate\[\int_0^1 \int_0^1 \cdots \int_0^1 f(x_1, x_2, \ldots, x_n) \, dx_1 \, dx_2 \cdots dx_n\]Use your result to calculate\[\sum_{n=0}^{\infty} \int_0^1 \int_0^1 \cdots \int_0^1 f(x_1, x_2, \ldots, x_n) \, dx_1 \, dx_2 \cdots dx_n\]**Problem 5:**The *average value* $ f_{\text{avg}} $ of the function $ f : \mathbb{R}^2 \to \mathbb{R} $ over the domain $ D $ is given by the formula\[f_{\text{avg}} = \frac{1}{m(D)} \iint_D f(x, y) \, dA\]where $ m(D) $ is the measure of the size of $ D $ (in general, this could be length, area, volume, etc.). Find the average value of the function $ f(x, y) = x \sin^2(xy) $ on the square $[0, \pi] \times [0, \pi]$.

Note: As per bartleby guidelines; for more than one question asked only the first one is to be answered. Answer for question 4: ∫01∫01....∫01fx1,x2,...,xndx1dx2.....dxn=∫01∫01....∫01x1x2....xn dx1dx2.....dxn=∫01∫01……∫01x122x2...xn 10dx2dx3....dxn=∫01∫01……∫0112x2...xn dx2dx3....dxn=12∫01∫01……∫01x22210x3...xn-1xn dx3....dxn-1dxn=122∫01∫01……∫01x3...xn-1xn dx3....dxn-1dxn=122∫01……∫01x32210x4...xn-1xn dx4....dxn=123∫01……∫01x42210x5...xn-1xn dx5....dxn=124∫01……∫01x5...xn-1xn dx5....dxn Do the same process for x5,x6,....,xn-1. Then, ∫01∫01....∫01fx1,x2,...,xndx1dx2.....dxn=12n-1∫01xndxn =12n-1xn2210 =12n-112 =12n To evaluate ∑n=0n=∞∫01∫01...∫01fx1,x2,...,xn dx1dx2...dxn: From the previous step, ∫01∫01...∫01fx1,x2,...,xn dx1dx2...dxn=12n Substitute for ∫01∫01...∫01fx1,x2,...,xn dx1dx2...dxn in ∑n=0n=∞∫01∫01...∫01fx1,x2,...,xn dx1dx2...dxn: ∑n=0n=∞∫01∫01...∫01fx1,x2,...,xn dx1dx2...dxn=∑n=0n=∞12n=120+∑n=1n=∞12n=1+1=2 Note: ∑n=1n=∞12n=1

Math

Advanced Math

Let f : R" → R be defined by f(x1, x2, ... , xn) = x1X2•••Xn on the cube [0, 1] × [0, 1] × · · × [0,1] (i.e. for 0 < x1 < 1,0 < x2 < 1,...,0 < ¤n < 1). Evaluate 1 ,Xn) dx1 dx2 · dxn X2, · ... .. •1 1 Use your result to calculate X2, ·.., Txp Ixp ("x dxn .. ... n=0

Let f : R" → R be defined by f(x1, x2, ... , xn) = x1X2•••Xn on the cube [0, 1] × [0, 1] × · · × [0,1] (i.e. for 0 < x1 < 1,0 < x2 < 1,...,0 < ¤n < 1). Evaluate 1 ,Xn) dx1 dx2 · dxn X2, · ... .. •1 1 Use your result to calculate X2, ·.., Txp Ixp ("x dxn .. ... n=0

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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Question

**Problem 4:**

Let $ f: \mathbb{R}^n \to \mathbb{R} $ be defined by $ f(x_1, x_2, \ldots, x_n) = x_1 x_2 \cdots x_n $ on the cube $[0, 1] \times [0, 1] \times \cdots \times [0, 1]$ (i.e. for $0 \leq x_1 \leq 1, 0 \leq x_2 \leq 1, \ldots, 0 \leq x_n \leq 1$). Evaluate

\[
\int_0^1 \int_0^1 \cdots \int_0^1 f(x_1, x_2, \ldots, x_n) \, dx_1 \, dx_2 \cdots dx_n
\]

Use your result to calculate

\[
\sum_{n=0}^{\infty} \int_0^1 \int_0^1 \cdots \int_0^1 f(x_1, x_2, \ldots, x_n) \, dx_1 \, dx_2 \cdots dx_n
\]

**Problem 5:**

The *average value* $ f_{\text{avg}} $ of the function $ f : \mathbb{R}^2 \to \mathbb{R} $ over the domain $ D $ is given by the formula

\[
f_{\text{avg}} = \frac{1}{m(D)} \iint_D f(x, y) \, dA
\]

where $ m(D) $ is the measure of the size of $ D $ (in general, this could be length, area, volume, etc.). Find the average value of the function $ f(x, y) = x \sin^2(xy) $ on the square $[0, \pi] \times [0, \pi]$.

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