Q.16 - Calculus

Question

Suppose $f$ is an integrable function $[a, b]$ and $k$ is a real number. Use pictures of Riemann sums to illustrate that the right sum for the function $k f\left(x\right)$ on $[a, b]$ is $k$ times the value of the right sum (with the same $n$ ) for $f$ on $[a, b]$. What happens as $n\to \infty$ ? What does this exercise say about the definite integrals${\int }_a^{b}f\left(x\right)dx$ and ${\int }_a^{b}kf\left(x\right)dx?$

Step-by-Step Solution

Answer

${\int }_0^2{x}^2 \mathrm{d}x$

Step 1 Given information

Suppose  is an integrable function  and  is a real number. Use pictures of Riemann sums to illustrate that the right sum for the function  on  is  times the value of the right sum (with the same  ) for  on . What happens as  ? What does this exercise say about the definite integrals and

Step 2 Calculation

The given region is under the function  and starts at  and ends at , so the required definite integral is: