Surface integrals of scalar functions over planes, spheres and cylinders, ∬_S f dS. 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
\[ \int\limits_{0}^{2 \pi}\int\limits_{0}^{\frac{\pi}{2}} \frac{27 \sin{\left(2 u \right)}}{2}\, du\, dv = 27 \pi \]
∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved
Answer \( 27 \pi \)
✓ 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
\[ \int\limits_{0}^{4}\int\limits_{0}^{2 \pi} 1\, du\, dv = 8 \pi \]
∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved
Answer \( 8 \pi \)
✓ 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
Evaluate \( \displaystyle \iint_S x y \, dS \), where \( \displaystyle S \) is the part of the plane \( \displaystyle x + y + z = 3 \) in the first octant.
Parametrize S by r(u, v) = <u, v, -u - v + 3>; dS = ‖r_u × r_v‖ du dv.
\[ \int\limits_{0}^{3}\int\limits_{0}^{3 - u} \sqrt{3} u v\, dv\, du = \frac{27 \sqrt{3}}{8} \]
∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved
Answer \( \frac{27 \sqrt{3}}{8} \)
✓ 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
✓ 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
Evaluate \( \displaystyle \iint_S 1 \, dS \), where \( \displaystyle S \) is the part of the plane \( \displaystyle x + y + z = 4 \) in the first octant.
Parametrize S by r(u, v) = <u, v, -u - v + 4>; dS = ‖r_u × r_v‖ du dv.
✓ 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
\[ \int\limits_{0}^{3}\int\limits_{0}^{2 \pi} v\, du\, dv = 9 \pi \]
∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved
Answer \( 9 \pi \)
✓ 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
\[ \int\limits_{0}^{2}\int\limits_{0}^{2 \pi} \left(2 v + 8 \cos^{2}{\left(u \right)}\right)\, du\, dv = 24 \pi \]
∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved
Answer \( 24 \pi \)
✓ 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
✓ 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
Evaluate \( \displaystyle \iint_S z \, dS \), where \( \displaystyle S \) is the part of the plane \( \displaystyle x + y + z = 1 \) in the first octant.
Parametrize S by r(u, v) = <u, v, -u - v + 1>; dS = ‖r_u × r_v‖ du dv.
\[ \int\limits_{0}^{1}\int\limits_{0}^{1 - u} \sqrt{3} \left(- u - v + 1\right)\, dv\, du = \frac{\sqrt{3}}{6} \]
∬ f(r(u, v)) ‖r_u × r_v‖ du dv.✓ Proved
Answer \( \frac{\sqrt{3}}{6} \)
✓ 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
Evaluate \( \displaystyle \iint_S x \, dS \), where \( \displaystyle S \) is the part of the plane \( \displaystyle x + y + z = 4 \) in the first octant.
Parametrize S by r(u, v) = <u, v, -u - v + 4>; dS = ‖r_u × r_v‖ du dv.
✓ 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