Find the average value of f(x, y) over the plane region R. f(x, y) = x R: rectangle with vertices (0, 0), (4, 0), (4, 2), (0, 2)

Find the average value of the function f(x, y) over the given region R. f(x, y) = 6xy; R is the triangle with vertices (0, 0), (0, 1), (3, 1).

Calculate the average value of the function over the given interval :

$$f(x)=\frac{1}{\sqrt{1-x^2}} ;\left[-\frac{1}{2}, 0\right]$$

Find the average value of the function f(x, y) over the given region R. $f(x, y)=\frac{y}{x}+\frac{x}{y};$ R: $1 \leq x \leq 4,1 \leq y \leq 3$

Find or approximate all points at which the given function equals its average value on the given interval.

$$f(x)=\frac{1}{x^{2}} \text { on }[1,4]$$

If a linear function passes through two points $(x_1, y_1)$ and $(x_2, y_2)$, what is the average value of the function on the interval from $x_1$ to $x_2$?

Does $f(x, y)=x$ have an average value over the region $0 \leq x<\infty, 0 \leq y \leq \frac{1}{1+x^{2}}$? If so, what is it?

Find the average value of the function f(x, y) over the given region R. $f(x, y)=\frac{\ln x}{x y};$ R: $1 \leq x \leq 2,2 \leq y \leq 3$

Find the average value of f(x, y) over the region R. f(x, y)=xy R: rectangle with vertices (0, 0), (4, 0), (4, 2), (0, 2)

Find the average value of f(x, y) over the plane region R. f(x, y) = x² + y² R: square with vertices (0, 0), (2, 0), (2, 2), (0, 2)

Determine whether the following statements are true and give an explanation or counterexample. Assume $f'$ and $f''$ are continuous functions for all real numbers.

If the average value of $f$ on [a, b] is zero then $f(x)=0$ on [a, b].

Find the average value of the function $f(x)=\frac{1}{\sqrt{9-4 x^{2}}}$ over the interval $\left[0,\frac{1}{2}\right]$.

Find the average value of f over the given rectangle. f(x, y) = x$^{2}$y, R has vertices (-1, 0), (-1, 5), (1, 5), (1, 0)

Does $f(x, y)=x y$ have an average value over the region $0 \leq x<\infty, 0 \leq y \leq \frac{1}{1+x^{2}}$? If so, what is it?