Find the area under the given curves and given lines: $(i) \: y = x^2, x = 1, x = 2 \: and\: x - axis$

This is first part of multipart q1

Toolbox:
• Area of a region bounded by the curve $y=f(x)$,$x$-axis and the lines $x=a,x=b$ is given by $A=\int_a^b dA=\int_a^b yda=\int_a^bf(x)dx.$
Step 1:
Given $f(x)=x^2,x=1,x=2$ ans $x$-axis
Using the information in the toolbox
we understand $A=\int_a^bf(x)dx$
Here $a=1$ and $b=2$ and $f(x)=x^2$
Therefore area $A=\int_1^2x^2dx$
$A=\begin{bmatrix}\large\frac{x^3}{3}\end{bmatrix}_1^2sq.units$
Step 2:
Applying the limits we get :
$A=\begin{bmatrix}\large\frac{2^3}{3}-\frac{1^3}{3}\end{bmatrix} sq.units.$
$\;\;\;=\begin{bmatrix}\large\frac{8}{3}-\frac{1}{3}\end{bmatrix}=\large\frac{7}{3}$sq.units
Hence the required area is $\large\frac{7}{3}$sq. units.