Question

Fill in the blanks: The definite integral of an integrable function f from $x=atox=b$is defined to be

where$∆x=\_\_\_\_\_,x_{k=}\_\_\_\_\_\_\_\_\&x_k^{*}=\_\_\_\_\_\_\_\_$

Short answer

The blanks are$\Delta x=\frac{b-a}{n},x_k=a+k\Delta x\&x_k^{*}=x_k$

Step-by-step solution

Step 1. Given information

Fill in the blanks given as$∆x=\_\_\_\_\_,x_{k=}\_\_\_\_\_\_\_\_\&x_k^{*}=\_\_\_\_\_\_\_\_$

Step 2. Fill the blanks

The definite integral can be written as

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

Therefore,$\Delta x=\frac{b-a}{n},x_k=a+k\Delta x\&x_k^{*}=x_k$