Real Analysis: Riemann Integration and Fundamental Theorem

Find the derivative with respect to $x$ of the integral from 0 to $x$ of the function $\sqrt{t^2 + 4} \, dt$.

Evaluate the integral of $\sqrt{t^3 + 5}$ from $x$ to 4 and find its derivative.

Evaluate the integral from 5 to $x^2$ of $\sqrt{t^3 - 4}$ and differentiate it using the chain rule.

Integrate the function from $x^2$ to $x^3$ where the function is $a0 t{t^4 - 2}$ and find its derivative applying the fundamental theorem.

Compute the derivative of the integral from 3 to x of $\sin(T^2)$ with respect to $x$.

Find the derivative of the integral from 0 to x of $\frac{T}{1 + T^3}$ with respect to $x$.

Evaluate the double integral $\int_{0}^{2} \int_{\sqrt[3]{x}}^{2} \frac{1}{y^4 + 1} \, dy \, dx$ by changing the order of integration.

Prove that the Dirichlet function, defined on the interval $[0, 1]$ as $f(x) = 1$ if $x$ is rational and $f(x) = 0$ if $x$ is irrational, is not Riemann integrable.

Show that the function $f(x) = x$ is Riemann integrable on the interval $[0, 1]$ and find its integral value.

For the Riemann integral, we always start with a partition of the x-axis. Then, we approximate the area under the function's graph by summing up the areas of rectangles formed by choosing points on the x-axis and using those points to determine the width and height of each rectangle. The main question is whether this method correctly represents the area, and we need to ensure that this method defines the integral accurately, even with different choices of partitions and values.

Let $f$ be the function defined on the closed interval from 0 to 1 by $f(x) = 0$ if $x$ is not a rational number, and $f(x) = 1$ if $x$ is a rational number. Determine if the function is Riemann integrable.

Exercise 1: Suppose you've got a function on the interval from 0 to 1 which is 0 except at finitely many points. Show that $f$ is integrable and the value of the integral is 0.

Exercise 2: Suppose $f$ is the function on the closed interval from 0 to 1 which is defined by $f(x) = 0$ if $x$ is not equal to $\frac{1}{n}$ for any natural number $n$ and $f(x) = 1$ if $x$ is equal to $\frac{1}{n}$ for some natural number $n$. Is this function Riemann integrable?

Given a function $f(x) = x$ on the interval $[1, 5]$, calculate the lower sum $L(f, P)$ and the upper sum $U(f, P)$ where $P = \{1, \frac{3}{2}, 2, 4, 5\}$.

Approximate the area under the curve of the function $f(x) = 1 + x^2$ on the interval $[-1, 1]$ using 4 subintervals. Determine the lower sum by choosing $x_i^*$ such that $f(x_i^*)$ is minimized within each subinterval, and the upper sum by choosing $x_i^*$ such that $f(x_i^*)$ is maximized within each subinterval.