Question

use geometry to evaluate the definite integral
int_{0}^{12} x d x
int_{0}^{12} x d x=square \text { (simplify your answer.) }

Explanation:

Step1: Analyze the geometric shape

The integral \(\int_{0}^{12}x\mathrm{d}x\) represents the area under the line \(y = x\) from \(x = 0\) to \(x=12\). The region is a right - triangle with base \(b = 12\) and height \(h=12\).

Step2: Calculate the area of the triangle

The formula for the area of a triangle is \(A=\frac{1}{2}\times base\times height\). Substituting \(b = 12\) and \(h = 12\) into the formula, we get \(A=\frac{1}{2}\times12\times12\).

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Answer:

\(72\)