Question

if ( g(x) ) is positive, then the integral ( int_{a}^{b} g(x) dx ) corresponds to the area beneath ( g(x) ) and above the ( x )-axis over the interval ( a, b ).
on ( 0, 8 ), the function ( g(x) ) is above the ( x )-axis and is therefore positive. thus, ( int_{0}^{8} g(x) dx ) equals the area of the triangle created by the function, the ( x )-axis, and the ( y )-axis.
this triangle is a right triangle with a side length of 8 along the ( x )-axis and a side length of 16 along the ( y )-axis. (give the numeric values.)
step 2
since the area of a triangle with base ( b ) and height ( h ) is ( a = \frac{1}{2} ), then our triangle has an area of. thus, ( int_{0}^{8} g(x) dx = 64 )

Explanation:

Step1: Recall the formula for the area of a triangle

The formula for the area of a triangle is \(A=\frac{1}{2}bh\), where \(b\) is the base and \(h\) is the height.

Step2: Identify the base and height

From the problem, the base \(b = 8\) (along the \(x -\)axis) and the height \(h=16\) (along the \(y -\)axis).

Step3: Calculate the area

Substitute \(b = 8\) and \(h = 16\) into the formula \(A=\frac{1}{2}bh\).

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Answer:

The area of the triangle is \(64\), so \(\int_{0}^{8}g(x)dx = 64\)