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Temperature-dependence refers to how specific properties of a material, such as its specific heat, change with temperature. The specific heat is essentially a measure of how much energy is needed to change the temperature of a substance. In the given problem, the specific heat, denoted by \( c \), is described as a function of temperature \( T \):

• \( c = 0.20 + 0.14T + 0.023T^2 \)

This formula indicates that as the temperature changes, the specific heat also varies based on the expression \(0.20 + 0.14T + 0.023T^2\).
Each term in this equation contributes to how specific heat adjusts with each increment in temperature:

• \( 0.20 \) is a constant term, signifying some baseline specific heat value.
• \( 0.14T \) represents the linear dependence on temperature.
• \( 0.023T^2 \) shows that as temperature increases, the specific heat increases in a nonlinear fashion.

This temperature-dependence is important for calculating energy changes accurately, as it demonstrates that more energy might be needed at higher temperatures than at lower ones.

Energy calculation in this context revolves around determining the amount of energy required to alter the temperature of a substance. We do this by integrating over the temperature range considering the mass and specific heat function.
In the problem, we use the formula for energy \( Q \):

• \( Q = \int_{T_1}^{T_2} m \cdot c(T) \cdot dT \)

Here, \( Q \) is the total energy, \( m \) is the mass of the substance (2.0 grams in this example), \( c(T) \) is the specific heat function dependent on temperature, and \( dT \) represents a small change in temperature. This formula essentially sums up the small energy contributions needed for each incremental temperature change from \( T_1 \) to \( T_2 \).
This integration process reflects that energy isn't added uniformly as temperature increases or decreases, but depends significantly on how specific heat varies with temperature.

The definite integral is a substantial part of solving this exercise, enabling us to compute the total energy precisely. The process involves integrating the specific heat function over the given temperature range.
For our calculation:

• Integral setup is: \[\int_{5}^{15} (0.20 + 0.14T + 0.023T^2) \, dT\]

This integral allows the determination of how much energy is required to increase the temperature of 2 grams of the substance from \(5^{\circ}\mathrm{C}\) to \(15^{\circ}\mathrm{C}\).To compute the definite integral, follow these steps:
1. **Integrate the function:** The result is: \[0.20T + 0.07T^2 + 0.00767T^3\]
2. **Evaluate from 5 to 15:** Substitute \(T = 15\) and \(T = 5\) into the integrated result and subtract.
3. **Calculate the difference:** The integral evaluates the total change, representing the total energy required in calories.
Thus, definite integrals in this way provide a complete solution by accumulating the "small pieces" added as energy over the interval \([5, 15]\). This method is vital for understanding not just the energy added, but also the impact of temperature-dependent properties on these energy needs.