Q. 5 - Calculus

Question

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

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

Step-by-Step Solution

Answer

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

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