Questions: Use geometry to evaluate the definite integral. [ int0^4 x d x ] ∫₀⁴ x dx = □ (Simplify your answer.)

Transcript text: Use geometry to evaluate the definite integral. \[ \int_{0}^{4} x d x \] $\int_{0}^{4} x d x=$ $\square$ (Simplify your answer.)

Solution

Solution Steps

Step 1: Set Up the Integral

We need to evaluate the definite integral: \[ \int_{0}^{4} x \, dx \]

Step 2: Interpret the Integral Geometrically

The integral represents the area under the curve $ y = x $ from $ x = 0 $ to $ x = 4 $. This area forms a right triangle with:

Base = 4
Height = 4

Step 3: Calculate the Area of the Triangle

The area $ A $ of a triangle is given by the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Substituting the values: \[ A = \frac{1}{2} \times 4 \times 4 = \frac{1}{2} \times 16 = 8 \]

Final Answer

Thus, the value of the definite integral is: \[ \boxed{8} \]

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