Asked Feb 23, 2019
Given the function f(x)=2x4, find the total area between f(x) and the x-axis over the interval [2,7].

Expert Answer

Step 1

f(x) = 2x - 4.

Point of intersection of f(x) and x axis will be given by value of x such that f(x) = 0

Hence, 2x-4=0, hence x = 4 / 2 = 2

Interval for area calculation is [-2,7]. So the point of intersection of f(x) and x-axis that is x=2 is inside this interval. So some portion of the area will be above x -axis and some portion below x-axis. 

Step 2

Please refer to the diagram on the white board. The shaded region is the area that we have to find. The interval starts from point A with x= -2 and ends at E with x = 7. The point of interesection of f(x) with x-axis is point C that we determined will have x = 2.

Thus the total area = |A1| + |A2| where

|A1| = absolute area under integral of f(x).dx over x = -2 to x = 2

and |A2| = absolute area under integral of f(x).dx over x = 2 to x= 7 

Step 3

Recall the famous rule of integration: Integral of xn = xn+1 / (n+1)



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