# What is the integral of the constant function f (x, y) = 5 over the rectangle [-2,3]\times[2,4]?

What is the integral of the constant function $f\left(x,y\right)=5$ over the rectangle $\left[-2,3\right]×\left[2,4\right]$?

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Step 1
Integration of a function gives the result as the area covered by the function with an appropriate axis. INtegrals are of two types, definite and indefinite integrals.
Definite integrals are defined for a limit of variable from a lower limit to an upper limit. This is represented by the expression:-
$I={\int }_{a}^{b}f\left(x\right)dx$
$=F\left(b\right)-F\left(a\right)$
Here, F(x) is the anti-derivative of the function f(x).
Step 2
The value of the definite integral is constructed by using the limits of x from -2 to 2, and the limts of y from 3 to 4 as $I={\int }_{3}^{4}{\int }_{-2}^{2}f\left(x,y\right)dxdy$. This integral is calculated as follows:-
$I={\int }_{3}^{4}{\int }_{-2}^{2}f\left(x,y\right)dxdy$
$={\int }_{3}^{4}{\int }_{-2}^{2}5dxdy$
$=5{\int }_{3}^{4}{\int }_{-2}^{2}dxdy$
$=5{\int }_{3}^{4}\left[x{\right]}_{-2}^{2}dy$
$=5{\int }_{3}^{4}\left(2-\left(-2\right)\right)dy$
$=20{\int }_{3}^{4}dy$
$=20\left[y{\right]}_{3}^{4}=20$