1. **Problem Statement:** Calculate the definite integral of the polynomial function $$P(\delta) = 2.2128\delta^{2} + 4.8508\delta - 0.0217$$ over the interval from 0 to 0.120. 2. **Formula Used:** The integral of a polynomial term $$a\delta^{n}$$ is $$\frac{a}{n+1}\delta^{n+1}$$. 3. **Integration Step:** Integrate each term of $$P(\delta)$$: $$\int P(\delta) d\delta = \int (2.2128\delta^{2} + 4.8508\delta - 0.0217) d\delta = \frac{2.2128}{3}\delta^{3} + \frac{4.8508}{2}\delta^{2} - 0.0217\delta + C$$ 4. **Apply Limits:** Evaluate the definite integral from 0 to 0.120: $$E_{1} = \left(\frac{2.2128}{3}\times 0.120^{3}\right) + \left(\frac{4.8508}{2}\times 0.120^{2}\right) - 0.0217 \times 0.120$$ 5. **Calculate Each Term:** - $$\frac{2.2128}{3} \times 0.120^{3} = 0.001274$$ - $$\frac{4.8508}{2} \times 0.120^{2} = 0.03493$$ - $$0.0217 \times 0.120 = 0.002604$$ 6. **Sum the Results:** $$E_{1} = 0.001274 + 0.03493 - 0.002604 = 0.03360$$ 7. **Final Answer:** $$\boxed{E_{1} = 0.03360 \text{ kN}\cdot\text{mm}}$$