# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- # **Key Concepts** This exercise focuses on the following concepts: - Definite integrals as signed area under curves - Geometric interpretation of definite integrals (rectangles, triangles, circles) - Properties of definite integrals (linearity and additivity) - Integration of piecewise-defined functions - Evaluating integrals using geometric area formulas - Interval additivity property: $\int_a^c = \int_a^b + \int_b^c$ --- # **Important Formulas** Below are the key formulas used in this exercise: **Linearity Property:** $$\int_{a}^{b} \left[ c_1 f(x) + c_2 g(x) \right] \, dx = c_1 \int_{a}^{b} f(x) \, dx + c_2 \int_{a}^{b} g(x) \, dx$$ **Interval Additivity:** $$\int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx$$ **Geometric Area Formulas:** | Shape | Formula | | :--- | :--- | | Rectangle | $A = \text{base} \times \text{height}$ | | Triangle | $A = \frac{1}{2} \times \text{base} \times \text{height}$ | | Circle | $A = \pi r^2$ | --- # **Summary** This exercise develops geometric and algebraic techniques for evaluating definite integrals without using antiderivatives. Key strategies include visualizing regions to calculate signed areas, applying linearity properties to combine integrals of functions, and using interval additivity to split integrals over adjacent domains. Special attention is given to piecewise functions (where integration must be performed separately over each subinterval) and recognizing standard geometric shapes (triangles, rectangles, semicircles) within integral bounds. The exercise emphasizes that definite integrals represent net signed area, requiring careful attention to whether regions lie above or below the x-axis.