The task is to graphically represent the following definite integrals from the image:

1. a) $\int_{-2}^{2} (-x + 4) \, dx$

   • This integral represents the area under the curve of the linear function $f(x) = -x + 4$ between $x = -2$ and $x = 2$.

2. b) $\int_{0}^{4} 3 \, dx$

   • This is the area under the constant function $f(x) = 3$ from $x = 0$ to $x = 4$, which will form a rectangle.

3. c) $\int_{1}^{3} (x^2 + 1) \, dx$

   • This integral represents the area under the curve $f(x) = x^2 + 1$, a parabolic curve, from $x = 1$ to $x = 3$.

4. d) $\int_{0.5}^{2} \frac{1}{x} \, dx$

   • This is the area under the curve $f(x) = \frac{1}{x}$, a hyperbolic function, between $x = 0.5$ and $x = 2$.

I can graph these functions and shade the areas corresponding to each definite integral. I will proceed by graphing all of them.It seems I encountered an issue while plotting the graph for part (b). This occurred because the function $f(x) = 3$ is a constant, and I tried applying it incorrectly across a range of $x$-values.

Let me correct the mistake and regenerate the plots for all the integrals correctly.It seems I forgot to redefine the variable $x_c$ when recreating the plots. Let me fix that and regenerate the graphs.