Q.0 - Calculus

Question

Read the section and make your own summary of the material. 

Step-by-Step Solution

Answer

The area under a curve, sigma notation and limits of sums can be reviewed 

Step 1. Summary

• Sigma Notation and Properties of sum.
• Limits of sum, and definition of definite integral.

A study of the accumulation function.

• The accumulation function A(x) gives the accumulation of area between the horizontal axis and the graph f(x) between 0 to x.

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

Step 2. Continue

• Sigma Notation and Properties of sum.

$$\begin{gathered} \sum _{k=\text{ starting\_value }}^{\text{ending\_value }} \left(\text{ function \_ of \_}k\right) \\ \text{ 1. }\sum _{k=1}^n \left(a_k+b_k\right)=\sum _{k=1}^n a_k+\sum _{k=1}^n b_k \\ \text{ 2. }\sum _{k=1}^n a_k=\sum _{k=1}^{p-1} a_k+\sum _{k=p}^n a_k \end{gathered}$$

• Calculating sums of areas of small rectangles using definite integral.

$\begin{array}{l}{\int }_0^x f\left(x\right)dx=\underset{n\to \mathrm{\infty }}{lim} \sum _{k=1}^n f\left(c_k\right)\mathrm{\Delta }x\\ \mathrm{\Delta }x=\frac{b-a}{n}\\ c_k=a+k\mathrm{\Delta }x\end{array}$

Thus, the area under a curve, sigma notation and limits of sums can be reviewed.