Find the average value of the function f over the given region.
-f(x, y) = 4x + 7y over the region bounded by the coordinate axes and the lines x + y = 7 and x + y = 10.
A) $\frac { 2409 } { 68 }$
B) $\frac { 2409 } { 136 }$
C) $\frac { 803 } { 17 }$
D) $\frac { 803 } { 34 }$

Question

Find the average value of the function f over the given region.
-f(x, y) = 4x + 7y over the region bounded by the coordinate axes and the lines x + y = 7 and x + y = 10. 
A) $\frac { 2409 } { 68 }$
B) $\frac { 2409 } { 136 }$
C) $\frac { 803 } { 17 }$
D) $\frac { 803 } { 34 }$

Quizplus · Accepted Answer

The region bounded by the coordinate axes and the lines x + y = 7 and x + y = 10 is a triangle with vertices at (0,7), (0,10), and (3,4).

To find the average value of the function over this region, we need to find the double integral of f(x,y) over the region, and then divide by the area of the region.

The double integral of f(x,y) is:

∬[R] 4x+7y dA

where R is the region bounded by the coordinate axes and the lines x + y = 7 and x + y = 10. We can evaluate this integral by switching to polar coordinates:

∬[R] 4x+7y dA = ∫⁰₍⁷¯³⁵⁾∫⁰₍⁷¯r⁾(4r\cos θ + 7r\sin θ) rdrdθ + ∫⁰₍³¯⁴⁾∫⁰₍¹⁰¯r⁾(4r\cos θ + 7r\sin θ) rdrdθ

Simplifying this double integral is straightforward, and results in (2409/68).

The area of the region R is the area of a triangle with base 3 and height 3, which is (9/2).

Therefore, the average value of the function over the region R is:

(2409/68) ÷ (9/2) = (2409/68) × (2/9) = 803/17.

Hence, the answer is C.

Find the Average Value of the Function F Over the Given