Question

Read the section and make your own summary of the material

Short answer

Rules for sums and multiples with constants for definite integrals

1. ${\int }_a^b\left[f\right(x)+g(x\left)\right]dx={\int }_a^{b}f\left(x\right)dx+{\int }_a^{b}g\left(x\right)dx$

2. ${\int }_a^{b}kf\left(x\right)dx=k{\int }_a^{b}f\left(x\right)dx$

Step-by-step solution

Given information

"DEFINITE INTEGRALS"

The precise region of a function's graph on $[0,x]$.

• Riemann's sums are defined as the limit of the definite integrals.
• Definite integral properties

Summary of "DEFINITE INTEGRALS".

A study of topics connected to definite integrals.

The area between the horizontal axis and the graph of $f\left(x\right)$from 0 to x is accumulated by the accumulation function $A\left(x\right)$.

$A\left(x\right)={\int }_0^{x}f\left(x\right)dx$

Utilizing definite integrals to calculate the sums of the areas of small rectangles.

${\int }_0^{x}f\left(x\right)dx=\underset{n\to \infty }{\lim }\sum _{k=1}^{n}f\left(c_k\right)\Delta x$

$$\begin{gathered} \Delta x=\frac{b-a}{n} \\ {c}_k=a+k\Delta x \end{gathered}$$

This makes it possible to determine the area under a curve.

This method of estimating the Riemann sums of the area between the curve and thex horizontal from 0 to x.

For definite Integrals, sum and constant multiple laws apply.

1. ${\int }_a^b\left[f\right(x)+g(x\left)\right]dx={\int }_a^{b}f\left(x\right)dx+{\int }_a^{b}g\left(x\right)dx$

2. ${\int }_a^{b}kf\left(x\right)dx=k{\int }_a^{b}f\left(x\right)dx$