QQuestionMathematics
QuestionMathematics
# QUESTION 2
Find the average value of the function $f(x, y)= 20 - 2 y$ over the rectangle $R=[0,3] \times[0,5]$.
$\bigcirc \mathrm{A} f_{\text {ave }}= 0$
$\bigcirc \mathrm{B} f_{\text {ave }}= 75$
$\bigcirc \mathrm{C} f_{\text {ave }}= 15$
$\bigcirc \mathrm{D} f_{\text {ave }}= 3$
$\bigcirc$ E None of these
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Answer
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Step 1:: Recall the formula for finding the average value of a function over a region.
where $A(R)$ is the area of the region $R$.
Step 2:: In this case, the function is $f(x, y) = 20 - 2y$ and the region $R$ is a rectangle with bounds $0 \leq x \leq 3$ and $0 \leq y \leq 5$.
A(R) = (3-0)(5-0) = 15
First, compute the area of the rectangle:
Step 3:: Now, compute the double integral of the function over the region $R$:
\iint_R f(x, y) dA = \int_0^5 \int_0^3 (20 - 2y) dx dy
Step 4:: Integrate with respect to $x$:
\int_0^3 (20x - 2xy) \Big|_0^3 dy = \int_0^5 (60 - 2y \cdot 3) dy
Step 5:: Integrate with respect to $y$:
\int_0^5 (60 - 6y) dy = \left[ 60y - 3y^2 \right]_0^5 = 300 - 75 = 225
Step 6:: Plug the results back into the formula:
f_{ave} = \frac{225}{15} = 15
Final Answer
The correct answer is $\bigcirc \mathrm{C}$.
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