Consider the function $f(x)$ whose graph is given below.
(a) Find $\int_{4}^{24} f(x) d x$.
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(b) What is the average value of $f$ on $[4,24]$ ?

Step 1 ：Let the function be $f(x) = x^2 - 3x + 2$. We want to find the integral of $f(x)$ from 4 to 24 and the average value of $f$ on [4, 24].

Step 2 ：First, find the integral of $f(x)$ from 4 to 24: $\int_{4}^{24} (x^2 - 3x + 2) dx$. The result is $\frac{11360}{3}$.

Step 3 ：Next, find the average value of $f$ on [4, 24] using the formula $average = \frac{1}{b - a} \int_{a}^{b} f(x) dx$. In this case, a = 4 and b = 24.

Step 4 ：Calculate the average value: $average = \frac{1}{24 - 4} \cdot \frac{11360}{3}$. The result is $\frac{568}{3}$.

Step 5 ：\(\boxed{\text{(a) The integral of } f(x) \text{ from 4 to 24 is } \frac{11360}{3}.}\)

Step 6 ：\(\boxed{\text{(b) The average value of } f \text{ on [4, 24] is } \frac{568}{3}.}\)