1. **Problem Statement:** Evaluate the definite integrals of the function $f(x)$ given its graph: a) $\int_{-2}^{3} f(x) \, dx$ b) $\int_{3}^{7} f(x) \, dx$ c) $\int_{7}^{1} f(x) \, dx$ 2. **Understanding the problem:** The integral $\int_a^b f(x) \, dx$ represents the net area between the graph of $f(x)$ and the x-axis from $x=a$ to $x=b$. Areas above the x-axis count as positive, and areas below count as negative. 3. **Step-by-step evaluation:** a) From $x=-2$ to $x=3$: - The graph is a straight line descending from near $(-3,6)$ to $(1,2)$, so from $-2$ to $1$ it is linear and above the x-axis. - From $x=1$ to $x=3$, the graph is part of a semicircular arc above the x-axis. - Calculate the area of the trapezoid from $x=-2$ to $x=1$: Heights at $x=-2$ and $x=1$ are approximately $f(-2) \approx 4$ and $f(1) = 2$. Area $= \frac{(4+2)}{2} \times (1 - (-2)) = 3 \times 3 = 9$ - Calculate the area under the semicircle from $x=1$ to $x=3$: The semicircle has radius $r=2$ (from $x=1$ to $x=5$), so half the semicircle from $1$ to $3$ is a quarter circle area. Area of full semicircle $= \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (2)^2 = 2\pi$ Area from $1$ to $3$ is half of that semicircle, so $= \pi$ - Total area from $-2$ to $3$ is $9 + \pi$. b) From $x=3$ to $x=7$: - From $3$ to $5$, the graph is the remaining half of the semicircle above the x-axis. - From $5$ to $7$, the graph declines linearly from $(5,2)$ to $(7,0)$ (interpolated), staying above the x-axis. - Area from $3$ to $5$ is the other half of the semicircle: $\pi$. - Area from $5$ to $7$ is a triangle with base $2$ and height $2$: Area $= \frac{1}{2} \times 2 \times 2 = 2$ - Total area from $3$ to $7$ is $\pi + 2$. c) From $x=7$ to $x=1$: - Integral limits are reversed, so $\int_7^1 f(x) \, dx = -\int_1^7 f(x) \, dx$. - Calculate $\int_1^7 f(x) \, dx$: From $1$ to $5$, semicircle area $= 2\pi$. From $5$ to $7$, triangle area $= 2$. Total $= 2\pi + 2$. - Therefore, $\int_7^1 f(x) \, dx = -(2\pi + 2) = -2\pi - 2$. 4. **Final answers:** a) $\int_{-2}^{3} f(x) \, dx = 9 + \pi$ b) $\int_{3}^{7} f(x) \, dx = \pi + 2$ c) $\int_{7}^{1} f(x) \, dx = -2\pi - 2$