# Find a formula for the area function of f(x) = 2x + 4 with lower limit a = 0.

Find a formula for the area function of f(x) = 2x + 4 with lower limit a = 0.
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Step 1
Integrals and derivatives are considered to be inverses of each other. For example if we integrate a function and then differentiate the integrated function, we will obtain the original function.
Integrals are used to calculate area under the curve by specifying the limits according to the intervals where the bounded area occurs. Whereas derivatives are used to test the continuity of the curve and find the critical points of the function.
Step 2
The given function is f(x)=2x+4.
The area under f(x) is given by integrating the function and specifying the lower and upper limits for the definite integral.
Lower limit of integral is given to be 0.
Let upper limit of the integral be t.
Area is given by:
$A={\int }_{0}^{t}f\left(x\right)dx$
$={\int }_{0}^{t}\left(2x+4\right)dx$
$={\left[{x}^{2}+4x\right]}_{0}^{t}$
$={t}^{2}+4t$
Hence, the formula for area under the curve is ${x}^{2}+4x$.