1. **Problem Statement:** Evaluate the definite integrals of the piecewise linear function $f(x)$ using the geometry of the graph. 2. **Given:** - $\int_4^{10} f(x) \, dx = 12$ - The graph is piecewise linear from $x=0$ to $x=10$. 3. **Approach:** Use the geometric areas under the curve to find the integrals. The integral of $f(x)$ over an interval corresponds to the net area between the curve and the $x$-axis. 4. **Step a) Calculate $\int_0^3 f(x) \, dx$:** - From $0$ to $1$, the function is $0$ (horizontal line at $y=0$), so area is $0$. - From $1$ to $3$, the graph forms a triangle with base $2$ and height $3$ (assuming from the description the slope rises to $3$ at $x=3$). - Area of triangle = $\frac{1}{2} \times 2 \times 3 = 3$. So, $$\int_0^3 f(x) \, dx = 0 + 3 = 3.$$ 5. **Step b) Calculate $\int_3^4 f(x) \, dx$:** - From $3$ to $4$, the graph forms a trapezoid or triangle. Assume height at $x=3$ is $3$ and at $x=4$ is $0$ (line drops to $0$). - Area of triangle = $\frac{1}{2} \times 1 \times 3 = 1.5$. So, $$\int_3^4 f(x) \, dx = 1.5.$$ 6. **Step c) Calculate $\int_0^4 f(x) \, dx$:** - Sum of areas from $0$ to $3$ and $3$ to $4$. - $$3 + 1.5 = 4.5.$$ 7. **Step d) Given $\int_4^{10} f(x) \, dx = 12$** 8. **Step e) Calculate $\int_0^{10} 6f(x) \, dx$:** - Use linearity of integrals: $$\int_0^{10} 6f(x) \, dx = 6 \times \int_0^{10} f(x) \, dx.$$ - Calculate $\int_0^{10} f(x) \, dx = \int_0^4 f(x) \, dx + \int_4^{10} f(x) \, dx = 4.5 + 12 = 16.5.$ - Multiply by 6: $$6 \times 16.5 = 99.$$ **Final answers:** - a) $3$ - b) $1.5$ - c) $4.5$ - d) $12$ - e) $99$