Question

Using integration, find the area of the region bounded by the line $y-1=x$, the x-axis, and the ordinates $x=-2$ and $x=3$

Answer 1

Hint: In this question, we first need to plot the given line within the given ordinates. Now, we need to find the area under the curve within the limits which can be calculated by integration. Now, on integrating the line in terms of x within the given ordinates gives the result..

Complete step-by-step answer:
Now, let us first plot the given line in the question within the ordinates

Here, we need to find the area of the shaded region.
As we already know from the applications of integration that
The space occupied by the curve along with the axis, under the given condition is called area of bounded region.
The area bounded by the curve $y=f\left( x \right)$ above the x-axis and between the lines $x=a,x=b$is given by
$\int\limits_{a}^{b}{y}dx=\int\limits_{a}^{b}{f\left( x \right)dx}$
Now, from the given line in the question we have
$\Rightarrow y-1=x$
Now, this can also be written as
$\Rightarrow y=x+1$
Now, the area of the shaded region can be found by integrating the above line equation within the given ordinates
$\Rightarrow \int\limits_{a}^{b}{y}dx$
Now, from the given conditions in the question we have
$y=x+1,a=-2,b=3$
Now, on substituting these respective values in the above formula we get,
$\Rightarrow \int\limits_{-2}^{3}{\left( x+1 \right)}dx$
As we already know from the properties of integration that
$\int{{{x}^{n}}dx}=\dfrac{{{x}^{n+1}}}{n+1}$
Now, on integrating the above expression using the formula we get,
$\Rightarrow \left( \dfrac{{{x}^{2}}}{2}+x \right)_{-2}^{3}$
Now, on substituting the values and simplifying it further we get,
$\Rightarrow \left( \dfrac{{{3}^{2}}}{2}+3-\left( \dfrac{{{\left( -2 \right)}^{2}}}{2}+\left( -2 \right) \right) \right)$
Now, his can be further written in the simplified form as
$\Rightarrow \left( \dfrac{9-4}{2}+3+2 \right)$
Now, on further simplification we get,
$\Rightarrow \dfrac{5}{2}+5$
$\Rightarrow 7.5$
Hence, the area of the region is 7.5

Note:Instead of integrating with respect to x in the given ordinates we can also calculate it by integrating it with respect to y and find the value of y for the given ordinate values to integrate within those limits. Both the methods give the same result.
It is important to note that while calculating we should not consider wrong limits or neglect any of the terms while integrating because neglecting any of the terms or calculation errors changes the result.