use geometry to evaluate the definite integral\n int_{0}^{12} x d x \n int_{0}^{12} x d x=square \text {…

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

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

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).

[ \begin{align*} A&=\frac{1}{2}\times12\times12\ &= 72 \end{align*} ]

Answer:

(72)