Solved: ?In the following exercises, the function f is given in terms of double

Chapter 5, Problem 51

(choose chapter or problem)

In the following exercises, the function f is given in terms of double integrals.

a. Determine the explicit form of the function f.

b. Find the volume of the solid under the surface z = f (x, y) and above the region R.

c. Find the average value of the function f on R.

d. Use a computer algebra system (CAS) to plot z = f (x, y) and \(z=f_{\text {ave }}\) in the same system of coordinates.

Show that if f and g are continuous on [a, b] and [c, d], respectively, then

\(\int_{a}^{b} \int_{c}^{d}[f(x)+g(y)] d y d x=(d-c) \int_{a}^{b} f(x) d x+\int_{a}^{b} \int_{c}^{d} g(y) d y d x=(b-a) \int_{c}^{d} g(y) d y+\int_{c}^{d} \int_{a}^{b} f(x) d x d y\)

Text Transcription:

z=f_text_ave

int_a^b_int_c^d[f(x)+g(y)]d.y.d_x=(d-c)int_a^b.f(x)d.x+int_a^b_int_c^d_g(y)d.y.d_x=(b-a)int_c^d.g(y)d.y+int_c^d_int_a^b_f(x)d.x.d.y