Calculus 2 Homework 1

If $x \sin (π x) = \int_{0}^{x^{2}} f (t) d t$ , where $f$ is a continuous function, find $f (4)$ .
Suppose $f$ is continuous, $f (0) = 0$ , $f (1) = 1$ , $f^{'} (x) > 0$ , and $\int_{0}^{1} f (x) d x = \frac{1}{3}$ . Find the value of the integral $\int_{0}^{1} f^{- 1} (y) d y$ .
If $\int_{0}^{4} e^{(x - 2)^{4}} d x = k$ , find the value of $\int_{0}^{4} x e^{(x - 2)^{4}} d x$ .
Evaluate $lim_{n \to \infty} (\frac{1}{\sqrt{n} \sqrt{n + 1}} + \frac{1}{\sqrt{n} \sqrt{n + 2}} + \dots + \frac{1}{\sqrt{n} \sqrt{n + n}}) .$

Algorithm Homework:

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).
Create a Python program that generates a random Riemann sum. This program should do the following:
1. Generate random partition points $x_{0}, x_{1}, x_{2}, \dots, x_{n}$ that satisfy the condition $max (x_{i + 1} - x_{i}) < δ$ for a given small positive value $δ$ , 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.

You can try your code here!