The average value of a function of two variables over a rectangle. 60 problems with worked solutions; in 60 of them every
equation is proved by a computer algebra system.
12 worked problems
Try each one before opening the solution. Every line is labeled with what checked it. How we verify
Find the average value of \( \displaystyle f(x, y) = x^{2} + y^{2} \) over \( \displaystyle R = [0, 1] \times [0, 2] \).
The average is (1/area) ∬_R f dA.
Reviewed
\[ \int\limits_{0}^{2}\int\limits_{0}^{1} \left(x^{2} + y^{2}\right)\, dx\, dy = \frac{10}{3} \]
The double integral.✓ Proved
\[ \frac{10}{1 \cdot 2 \cdot 3} = \frac{5}{3} \]
Divide by the area 2.✓ Proved
Answer \( \frac{5}{3} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the average value of \( \displaystyle f(x, y) = x y \) over \( \displaystyle R = [0, 2] \times [0, 1] \).
The average is (1/area) ∬_R f dA.
Reviewed
\[ \int\limits_{0}^{1}\int\limits_{0}^{2} x y\, dx\, dy = 1 \]
The double integral.✓ Proved
\[ 1 \frac{1}{1 \cdot 2} = \frac{1}{2} \]
Divide by the area 2.✓ Proved
Answer \( \frac{1}{2} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the average value of \( \displaystyle f(x, y) = - x + 2 y \) over \( \displaystyle R = [0, 1] \times [0, 3] \).
The average is (1/area) ∬_R f dA.
Reviewed
\[ \int\limits_{0}^{3}\int\limits_{0}^{1} \left(- x + 2 y\right)\, dx\, dy = \frac{15}{2} \]
The double integral.✓ Proved
\[ \frac{15}{1 \cdot 2 \cdot 3} = \frac{5}{2} \]
Divide by the area 3.✓ Proved
Answer \( \frac{5}{2} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the average value of \( \displaystyle f(x, y) = 3 x + y^{2} \) over \( \displaystyle R = [1, 3] \times [0, 1] \).
The average is (1/area) ∬_R f dA.
Reviewed
\[ \int\limits_{0}^{1}\int\limits_{1}^{3} \left(3 x + y^{2}\right)\, dx\, dy = \frac{38}{3} \]
The double integral.✓ Proved
\[ \frac{38}{1 \cdot 2 \cdot 3} = \frac{19}{3} \]
Divide by the area 2.✓ Proved
Answer \( \frac{19}{3} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the average value of \( \displaystyle f(x, y) = x^{2} y \) over \( \displaystyle R = [0, 2] \times [0, 3] \).
The average is (1/area) ∬_R f dA.
Reviewed
\[ \int\limits_{0}^{3}\int\limits_{0}^{2} x^{2} y\, dx\, dy = 12 \]
The double integral.✓ Proved
\[ \frac{12}{2 \cdot 3} = 2 \]
Divide by the area 6.✓ Proved
Answer \( 2 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the average value of \( \displaystyle f(x, y) = 3 x + y^{2} \) over \( \displaystyle R = [-1, 1] \times [0, 3] \).
The average is (1/area) ∬_R f dA.
Reviewed
\[ \int\limits_{0}^{3}\int\limits_{-1}^{1} \left(3 x + y^{2}\right)\, dx\, dy = 18 \]
The double integral.✓ Proved
\[ \frac{18}{2 \cdot 3} = 3 \]
Divide by the area 6.✓ Proved
Answer \( 3 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the average value of \( \displaystyle f(x, y) = - x + 2 y \) over \( \displaystyle R = [-1, 1] \times [0, 3] \).
The average is (1/area) ∬_R f dA.
Reviewed
\[ \int\limits_{0}^{3}\int\limits_{-1}^{1} \left(- x + 2 y\right)\, dx\, dy = 18 \]
The double integral.✓ Proved
\[ \frac{18}{2 \cdot 3} = 3 \]
Divide by the area 6.✓ Proved
Answer \( 3 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the average value of \( \displaystyle f(x, y) = x y \) over \( \displaystyle R = [-1, 1] \times [0, 1] \).
The average is (1/area) ∬_R f dA.
Reviewed
\[ \int\limits_{0}^{1}\int\limits_{-1}^{1} x y\, dx\, dy = 0 \]
The double integral.✓ Proved
\[ \frac{0}{1 \cdot 2} = 0 \]
Divide by the area 2.✓ Proved
Answer \( 0 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the average value of \( \displaystyle f(x, y) = x^{2} + y^{2} \) over \( \displaystyle R = [-1, 1] \times [0, 1] \).
The average is (1/area) ∬_R f dA.
Reviewed
\[ \int\limits_{0}^{1}\int\limits_{-1}^{1} \left(x^{2} + y^{2}\right)\, dx\, dy = \frac{4}{3} \]
The double integral.✓ Proved
\[ \frac{4}{1 \cdot 2 \cdot 3} = \frac{2}{3} \]
Divide by the area 2.✓ Proved
Answer \( \frac{2}{3} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find the average value of \( \displaystyle f(x, y) = x^{2} y \) over \( \displaystyle R = [-1, 1] \times [0, 2] \).
The average is (1/area) ∬_R f dA.
Reviewed
\[ \int\limits_{0}^{2}\int\limits_{-1}^{1} x^{2} y\, dx\, dy = \frac{4}{3} \]
The double integral.✓ Proved
\[ \frac{4}{2 \cdot 2 \cdot 3} = \frac{1}{3} \]
Divide by the area 4.✓ Proved
Answer \( \frac{1}{3} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify