Question

1. no calculator the graph of ( y = f(x) ) is shown in the figure above. if ( a_1 ) and ( a_2 ) are positive numbers that represent the areas of the shaded regions, then in terms of ( a_1 ) and ( a_2 ), ( int_{-4}^{4} f(x) dx - 2 int_{-1}^{4} f(x) dx = ) (a) ( a_1 ) (b) ( a_1 - a_2 ) (c) ( 2a_1 - a_2 ) (d) ( a_1 + a_2 ) (e) ( a_1 + 2a_2 )

Explanation:

Step1: Analyze the definite - integral based on the area above and below the \(x\) - axis

Recall that if \(y = f(x)\) and \(A_1\) is the area above the \(x\) - axis and \(A_2\) is the area below the \(x\) - axis, then \(\int_{-4}^{4}f(x)dx=\int_{-4}^{-1}f(x)dx+\int_{-1}^{4}f(x)dx\). The integral \(\int_{a}^{b}f(x)dx\) is equal to \(A\) (the area between \(y = f(x)\), \(x=a\), \(x = b\) and the \(x\) - axis) when \(f(x)\geq0\) on \([a,b]\) and \(-A\) when \(f(x)\leq0\) on \([a,b]\). So \(\int_{-4}^{-1}f(x)dx=A_1\) and \(\int_{-1}^{4}f(x)dx=-A_2\). Then \(\int_{-4}^{4}f(x)dx=A_1 - A_2\).

Step2: Analyze \(\int_{-1}^{4}f(x)dx\)

We know that \(\int_{-1}^{4}f(x)dx=-A_2\).

Step3: Calculate \(\int_{-4}^{4}f(x)dx-2\int_{-1}^{4}f(x)dx\)

Substitute the values of the integrals:

$$
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Answer:

D. \(A_1 + A_2\)