Question

How do you use the limit process to find the area of the region between the graph $y = {x^2} + 1$ and the $x$ axis over the interval $\left[ {0,3} \right]$ ?

Answer 1

Here we will use integration to find the area of the region bounded on the certain limit. The area $A$ of the region under the graph of a function $f(x) \geqslant 0$ above the $x$ axis from $x = a$ and $x = b$ can be found by $A = \int\limits_a^b {f(x)\,dx}$ .
Formula used:
If $f(x) = {x^n}$ then $\int {{x^n}\,dx} = \dfrac{{{x^{n + 1}}}}{{n + 1}} + C$

Complete step by step answer: By the definition of an integral
$A = \int\limits_a^b {f(x)\,dx}$ represents the area under the curve $y = f(x)$ between $x = a$ and $x = b$.
Now we have the function $y = {x^2} + 1$ and the interval $\left[ {0,3} \right]$
Let write down in integration form, that is,
$\Rightarrow A = \int\limits_0^3 {{x^2} + 1\,dx}$
Here the upper limit value is $x = 0$and lower limit value is $x = 3$
Now using the integration property,
$\int {f(x) + g(x)\,dx} = \int {f(x)\,dx + \int {g(x)\,dx} }$
By the above property separate the integration,
$\Rightarrow A = \int\limits_0^3 {{x^2}\,dx} + \int\limits_0^3 {1\,dx}$
Now using the integration formula that is,
$\int {{x^n}\,dx} = \dfrac{{{x^{n + 1}}}}{{n + 1}} + C$
Now we get the result,
$\Rightarrow A = \left[ {\dfrac{{{x^{2 + 1}}}}{{2 + 1}}} \right]_0^3 + \left[ x \right]_0^3$
On simplified,
$\Rightarrow A = \left[ {\dfrac{{{x^3}}}{3}} \right]_0^3 + \left[ x \right]_0^3$
Now applying the upper limit and the lower limit, we know that while applying the limit we must follow the rule that is we first apply the upper limit value to the variable $x$ and then subtract the resulting value with the lower limit value which is applied to the variable $x$.
$\;\;$ $\Rightarrow A = \left[ {\dfrac{{{3^3}}}{3} - 0} \right] + \left[ {3 - 0} \right]$
Simplify the above we get,
$\Rightarrow A = \left[ {{3^2}} \right] + \left[ 3 \right]$
On square of the required number,
$\Rightarrow A = 9 + 3$
By adding,
$\Rightarrow A = 12$
Hence the area of the region of a given function bounded within the interval is $12$.

Note:
The area under a curve between two points can be found by doing a definite integral between the two points.
To find the area of the region $y = f(x)$ between $x = a$ and $x = b$ , integrate $y = f(x)$ between the limits of $a$ and $b$.
Areas under the $x$ axis will come out negative and areas above the $x$ axis will be positive.
This means that you have to be careful when finding an area which is partly above and partly below the $x$ axis.