1. **Problem Statement:** Find the definite integrals of the function $y=J(x)$ over the given intervals: (a) $\int_{-8}^{4} f(x) \, dx$ (b) $\int_{-8}^{0} f(x) \, dx$ (c) $\int_{3}^{0} f(x) \, dx$ (d) $\int_{2}^{5} f(x) \, dx$ (e) $\int_{5}^{7} f(x) \, dx$ 2. **Understanding the graph and function:** The function is piecewise with segments: - $[-8,-6]$: line from $( -8,0 )$ to $( -6,-2 )$ - $[-6,-3]$: V shape with vertex at $(-4,3)$ - $[-3,0]$: semicircle above x-axis with radius 3 - $[0,4]$: horizontal line at $y=1$ - $[4,6]$: line at $y=0$ - $[6,7]$: line from $(6,0)$ to $(7,-1)$ 3. **Calculate areas for each segment:** (a) $\int_{-8}^{4} f(x) \, dx$ includes segments from $-8$ to $4$: - From $-8$ to $-6$: trapezoid area - From $-6$ to $-3$: V shape area - From $-3$ to $0$: semicircle area - From $0$ to $4$: rectangle area Calculate each: - $[-8,-6]$ line area: Base length = $2$ Heights = $0$ and $-2$ Area = $\frac{1}{2} \times 2 \times (0 + (-2)) = -2$ - $[-6,-3]$ V shape: Two triangles each with base $2$ and height $3$ Area = $2 \times \frac{1}{2} \times 2 \times 3 = 6$ - $[-3,0]$ semicircle: Radius $r=3$ Area = $\frac{1}{2} \pi r^2 = \frac{1}{2} \pi \times 9 = \frac{9\pi}{2}$ - $[0,4]$ rectangle: Height = $1$, width = $4$ Area = $4$ Sum all: $$-2 + 6 + \frac{9\pi}{2} + 4 = 8 + \frac{9\pi}{2}$$ 4. (b) $\int_{-8}^{0} f(x) \, dx$ includes $[-8,-6]$, $[-6,-3]$, $[-3,0]$: Sum areas: $$-2 + 6 + \frac{9\pi}{2} = 4 + \frac{9\pi}{2}$$ 5. (c) $\int_{3}^{0} f(x) \, dx$ is integral from $0$ to $3$ reversed: From $0$ to $3$, $y=1$ Area = $3$ Reverse limits: $$\int_{3}^{0} f(x) \, dx = - \int_{0}^{3} 1 \, dx = -3$$ 6. (d) $\int_{2}^{5} f(x) \, dx$ includes $[2,4]$ at $y=1$ and $[4,5]$ at $y=0$: - $[2,4]$: width $2$, height $1$, area $2$ - $[4,5]$: height $0$, area $0$ Sum: $$2 + 0 = 2$$ 7. (e) $\int_{5}^{7} f(x) \, dx$ includes $[5,6]$ at $y=0$ and $[6,7]$ line down to $-1$: - $[5,6]$: area $0$ - $[6,7]$: triangle with base $1$, height $-1$ Area = $\frac{1}{2} \times 1 \times (-1) = -\frac{1}{2}$ Sum: $$0 - \frac{1}{2} = -\frac{1}{2}$$ **Final answers:** (a) $8 + \frac{9\pi}{2}$ (b) $4 + \frac{9\pi}{2}$ (c) $-3$ (d) $2$ (e) $-\frac{1}{2}$