**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