1. **Problem Statement:** Calculate the definite integrals of the piecewise function $f(x)$ over the intervals $[-1,2]$, $[2,4]$, and $[-1,7]$ based on the graph description. 2. **Understanding the function:** - From $x=-2$ to $x=0$, $f(x)$ is linear and decreasing. - From $x=0$ to $x=3$, $f(x)$ forms a semicircle with lowest $y=-3$ and highest near $0$. - From $x=3$ to $x=7$, $f(x)$ increases linearly from $0$ to $2$. 3. **Integral over $[-1,2]$:** - From $x=-1$ to $x=0$, $f(x)$ decreases linearly from $-1$ to $-3$. - From $x=0$ to $x=2$, $f(x)$ is part of the semicircle. Calculate area from $-1$ to $0$: This is a trapezoid with bases $|-1|$ and $|-3|$ and height $1$. Area = $\frac{1}{2} (|-1| + |-3|) \times 1 = \frac{1}{2} (1 + 3) = 2$ but since $f(x)$ is negative, integral is $-2$. Calculate area from $0$ to $2$: The semicircle radius is $1.5$ (since from $0$ to $3$), so area of full semicircle is $\frac{1}{2} \pi (1.5)^2 = \frac{1}{2} \pi \times 2.25 = 1.125\pi$. From $0$ to $2$ is $\frac{2}{3}$ of the semicircle, so area = $\frac{2}{3} \times 1.125\pi = 0.75\pi$. Since semicircle is below $x$-axis, integral is $-0.75\pi$. Sum for $[-1,2]$: $$\int_{-1}^2 f(x) dx = -2 - 0.75\pi$$ 4. **Integral over $[2,4]$:** - From $2$ to $3$, semicircle continues (last $1$ unit of semicircle). - From $3$ to $4$, linear increase from $0$ to about $0.5$ (since from $3$ to $7$ it goes from $0$ to $2$, slope $= \frac{2-0}{7-3} = 0.5$ per unit). Area semicircle from $2$ to $3$: This is $\frac{1}{3}$ of the semicircle area: $\frac{1}{3} \times 1.125\pi = 0.375\pi$ (negative area). Area from $3$ to $4$: Trapezoid with bases $0$ and $0.5$, height $1$. Area = $\frac{1}{2} (0 + 0.5) \times 1 = 0.25$ (positive). Sum for $[2,4]$: $$\int_2^4 f(x) dx = -0.375\pi + 0.25$$ 5. **Integral over $[-1,7]$:** Sum integrals over $[-1,2]$, $[2,4]$, and $[4,7]$. From $4$ to $7$: Linear increase from $0.5$ at $x=4$ to $2$ at $x=7$. Average height = $\frac{0.5 + 2}{2} = 1.25$, width = $3$. Area = $1.25 \times 3 = 3.75$. Sum all: $$\int_{-1}^7 f(x) dx = (-2 - 0.75\pi) + (-0.375\pi + 0.25) + 3.75 = ( -2 + 0.25 + 3.75 ) - (0.75\pi + 0.375\pi) = 2 - 1.125\pi$$ **Final answers:** $$\int_{-1}^2 f(x) dx = -2 - 0.75\pi$$ $$\int_2^4 f(x) dx = -0.375\pi + 0.25$$ $$\int_{-1}^7 f(x) dx = 2 - 1.125\pi$$