The Definite Integral as the Limit of a Sum

Divide the interval $[-1,4]$ into $n$ equal subintervals, and take the coordinates of the midpoints of these subintervals as the values of the independent variable $\xi_{i}$ ($i=0,1,\ldots ,n-1$). Find the integral sum $S_{n}$ of the function $f(x)=1+x$ over this interval.

Let

Divide the closed interval $[1,2]$ into $n$ parts such that the abscissas of the dividing points form a geometric series, to find the lower integral sum of the function $f(x)=x^{4}$ on $[1,2]$. What does the limit of this sum equal as $n\to \infty$?

Starting from the definition of the integral, find

where $v_{0}$ and $g$ are constants.

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum to compute the definite integral: $\displaystyle \int_{-1}^{2}x^{2}\, dx$.

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum to compute the definite integral: $\displaystyle \int_{0}^{1}a^{x}\, dx$ ($a>0$).

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum to compute the definite integral: $\displaystyle \int_{0}^{\frac{\pi }{2}}\sin x\, dx$.

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum to compute the definite integral: $\displaystyle \int_{0}^{x}\cos t\, dt$.

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum to compute the definite integral: $\displaystyle \int_{a}^{b}\dfrac {dx}{x^{2}}$ ($0<a<b$).

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum to compute the definite integral: $\displaystyle \int_{a}^{b}x^{m}\, dx$ ($0<a<b$; $m\neq -1$).

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum to compute the definite integral: $\displaystyle \int_{a}^{b}\dfrac {dx}{x}$ ($0<a<b$).

Using a suitable method of subdividing the interval of integration, regard the integral as the limit of the corresponding integral sum, compute the Poisson integral

when:

Suppose the functions $f(x)$ and $\varphi (x)$ are continuous on $[a,b]$, prove that

where $x_{i}\leqslant \xi_{i}\leqslant x_{i+1}$, $x_{i}\leqslant \theta_{i}\leqslant x_{i+1}$ ($i=0,1,\ldots ,n-1$) and $\Delta x_{i}=x_{i+1}-x_{i}$ ($x_{0}=a$, $x_{n}=b$).

Prove that the discontinuous function:

is integrable on the interval $[0,1]$.

Prove that the Riemann function

(where $m$ and $n$ ($n\geqslant 1$) are coprime integers) is integrable on any finite interval.

Prove that the function

is integrable on the closed interval $[0,1]$.

Prove that the Dirichlet function

is not integrable on any interval.

Suppose the function $f(x)$ is integrable on $[a,b]$, and

where $x_{i}=a+\dfrac {i}{n}(b-a)$ ($i=0,1,\ldots ,n-1$; $n=1,2,\ldots$).

Prove that $\displaystyle \lim_{n\to \infty }\int_{a}^{b}f_{n}(x)\, dx=\int_{a}^{b}f(x)\, dx$.

Prove that if the function $f(x)$ is integrable on $[a,b]$, then there is a sequence of continuous functions $\varphi_{n}(x)$ ($n=1,2,\ldots$) such that

Prove that if the bounded function $f(x)$ is integrable on the closed interval $[a,b]$, then its absolute value $\left|f(x)\right|$ is also integrable on $[a,b]$, and

If the function $f(x)$ is absolutely integrable on the closed interval $[a,b]$ (that is, the integral $\displaystyle \int_{a}^{b}\left|f(x)\right|\, dx$ exists), is this function necessarily integrable on $[a,b]$?

Let the function $\varphi (x)$ be defined and continuous on the closed interval $[A,B]$, let the function $f(x)$ be integrable on $[a,b]$, and let $A\leqslant f(x)\leqslant B$ when $a\leqslant x\leqslant b$. Prove that the function $\varphi [f(x)]$ is integrable on $[a,b]$.

If the functions $f(x)$ and $\varphi (x)$ are integrable, must the function $f[\varphi (x)]$ also be integrable?

Let the function $f(x)$ be integrable on the closed interval $[A,B]$. Prove that the function $f(x)$ has continuity in the mean of the integral, that is,

where $[a,b]\subset [A,B]$.

Let the function $f(x)$ be integrable on the closed interval $[a,b]$. Prove that the equality

holds if and only if $f(x)=0$ at every point of the closed interval $[a,b]$ where $f(x)$ is continuous.