Algorithm Homework:

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If $x\sin(\pi x) = \int_0^{x^2}f(t)dt$, where $f$ is a continuous function, find $f(4)$.

Suppose $f$ is continuous, $f(0)=0$, $f(1)=1$, $f’(x)>0$, and $\int_0^1f(x)dx=\frac{1}{3}$. Find the value of the integral $\int_0^1f^{-1}(y)dy$.

If $\int_0^4e^{(x-2)^4}dx=k$, find the value of $\int_0^4xe^{(x-2)^4}dx$.

Evaluate \(\lim_{n\to\infty}\left(\frac{1}{\sqrt{n}\sqrt{n+1}}+\frac{1}{\sqrt{n}\sqrt{n+2}}+\cdots+\frac{1}{\sqrt{n}\sqrt{n+n}}\right).\)

  1. Implement a Python program that calculates the left, right, and midpoint Riemann sums for a given function $f(x)$ over a specified interval $[a, b]$. Your program should allow the user to input the function, the interval, and the number of subintervals ($n$) they want to use for the Riemann sum. It should then compute and display the results for each type of Riemann sum (left, right, and midpoint).

  2. Create a Python program that generates a random Riemann sum. This program should do the following:

    1. Generate random partition points $x_0, x_1, x_2, \ldots, x_n$ that satisfy the condition $\max(x_{i+1} - x_i) < \delta$ for a given small positive value $\delta$, which also allows the user to select.
    2. Randomly select sample points $x_i^*$ within each subinterval.
    3. Evaluate the Riemann sum using these randomly generated values and display the result.