Questions: Use geometry to evaluate the definite integral. [ int0^11 x dx ]

Solution Steps

The integral \(\int_{0}^{11} x \, dx\) represents the area under the line \(y = x\) from \(x = 0\) to \(x = 11\). This area forms a right triangle with the x-axis. The base of the triangle is 11 and the height is also 11.

We need to evaluate the definite integral: \[ \int_{0}^{11} x \, dx \] This integral represents the area under the curve \(y = x\) from \(x = 0\) to \(x = 11\).

The area under the curve from \(x = 0\) to \(x = 11\) forms a right triangle with:

• Base = 11 (the length along the x-axis)
• Height = 11 (the value of the function at \(x = 11\))

The area \(A\) of a triangle is given by the formula: \[ A = \frac{1}{2} \times \text{base} \times \text{height} \] Substituting the values: \[ A = \frac{1}{2} \times 11 \times 11 = \frac{121}{2} = 60.5 \]

Final Answer