1. **Problem statement:** Evaluate the definite integrals of the piecewise linear function $f(x)$ given by the graph: - From $x=0$ to $x=2$, $f(x)=0$. - From $x=2$ to $x=3$, $f(x)$ rises linearly from 0 to 4. - From $x=3$ to $x=10$, $f(x)$ rises linearly from 4 to 8. 2. **Formula and approach:** The definite integral of $f(x)$ over an interval is the net area under the curve between those $x$-values. - For linear segments, the area can be found using geometry (triangles and trapezoids). 3. **Calculate each integral:** **a) $\int_0^3 f(x) \, dx$** - From 0 to 2, $f(x)=0$, so area is 0. - From 2 to 3, the graph forms a triangle with base $1$ and height $4$. - Area of triangle = $\frac{1}{2} \times 1 \times 4 = 2$. - Total area = $0 + 2 = 2$. **b) $\int_3^4 f(x) \, dx$** - From 3 to 4, $f(x)$ rises linearly from 4 to a value on the line from (3,4) to (10,8). - Slope from 3 to 10 is $\frac{8-4}{10-3} = \frac{4}{7}$. - At $x=4$, $f(4) = 4 + \frac{4}{7} \times (4-3) = 4 + \frac{4}{7} = \frac{32}{7}$. - Area from 3 to 4 is a trapezoid with bases 4 and $\frac{32}{7}$ and height 1. - Area = $\frac{1}{2} (4 + \frac{32}{7}) \times 1 = \frac{1}{2} \times \frac{60}{7} = \frac{30}{7} \approx 4.2857$. **c) $\int_0^4 f(x) \, dx$** - Sum of parts a and b: $2 + \frac{30}{7} = \frac{14}{7} + \frac{30}{7} = \frac{44}{7} \approx 6.2857$. **d) $\int_4^{10} f(x) \, dx$** - From 4 to 10, $f(x)$ rises linearly from $\frac{32}{7}$ to 8. - Area is trapezoid with bases $\frac{32}{7}$ and 8, height 6. - Area = $\frac{1}{2} (\frac{32}{7} + 8) \times 6 = 3 \times (\frac{32}{7} + 8) = 3 \times \frac{32 + 56}{7} = 3 \times \frac{88}{7} = \frac{264}{7} \approx 37.7143$. **e) $\int_0^{10} -4 f(x) \, dx$** - First find $\int_0^{10} f(x) \, dx$. - From 0 to 2, area = 0. - From 2 to 3, triangle area = 2. - From 3 to 10, trapezoid area with bases 4 and 8, height 7. - Area = $\frac{1}{2} (4 + 8) \times 7 = 6 \times 7 = 42$. - Total area $= 0 + 2 + 42 = 44$. - Multiply by -4: $-4 \times 44 = -176$. 4. **Final answers:** - a) $2$ - b) $\frac{30}{7} \approx 4.2857$ - c) $\frac{44}{7} \approx 6.2857$ - d) $\frac{264}{7} \approx 37.7143$ - e) $-176$ These results use geometric area calculations of triangles and trapezoids under the piecewise linear graph.