The volume of the right circular cylinder with a base of radius 4 in the ry-plane and height 5 is 5 27 c4 (a) I drdedz. (b) rdrdodz . drdOdz . Jo Jo The volume of the region inside the cone o = t/4 for 0
The volume of the right circular cylinder with a base of radius 4 in the ry-plane and height 5 is 5 27 c4 (a) I drdedz. (b) rdrdodz . drdOdz . Jo Jo The volume of the region inside the cone o = t/4 for 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Concept explainers
Cylinders
A cylinder is a three-dimensional solid shape with two parallel and congruent circular bases, joined by a curved surface at a fixed distance. A cylinder has an infinite curvilinear surface.
Cones
A cone is a three-dimensional solid shape having a flat base and a pointed edge at the top. The flat base of the cone tapers smoothly to form the pointed edge known as the apex. The flat base of the cone can either be circular or elliptical. A cone is drawn by joining the apex to all points on the base, using segments, lines, or half-lines, provided that the apex and the base both are in different planes.
Question
100%
![**The Volume of Geometric Shapes Using Integration**
1. **Volume of the Right Circular Cylinder**
To find the volume of a right circular cylinder with a base of radius 4 in the xy-plane and height 5, evaluate the following integrals:
- **Option (a):**
\[
\int_0^5 \int_0^{2\pi} \int_0^4 dr\,d\theta\,dz
\]
- **Option (b):**
\[
\int_0^5 \int_0^{2\pi} r\,dr\,d\theta\,dz
\]
- **Option (c):**
\[
\int_0^5 \int_0^{2\pi} \int_0^2 dr\,d\theta\,dz
\]
2. **Volume Inside the Cone \(\phi = \pi/4\), \(0 \leq z \leq 4\)**
To find the volume of the region inside the cone where \(\phi = \pi/4\) and \(0 \leq z \leq 4\), evaluate these integrals:
- **Option (a):**
\[
\int_0^{2\pi} \int_0^{\pi/4} \int_0^{4\sqrt{2}} \rho\,d\rho\,d\phi\,d\theta
\]
- **Option (b):**
\[
\int_0^{2\pi} \int_0^{\pi/4} \int_0^4 \rho^2 \sin\phi\,d\rho\,d\phi\,d\theta
\]
- **Option (c):**
\[
\int_0^{2\pi} \int_0^{\pi/4} \int_0^{4\sec\theta} \rho^2 \sin\phi\,d\rho\,d\phi\,d\theta
\]
These integrals are used to calculate the volumes based on the specific geometrical constraints and limits provided in the integrals.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25fb0993-5d26-4149-a829-39bf7f0bd1d0%2Fc76e05cb-9c1a-4ec8-9567-d14c43ab7418%2F8f0srz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**The Volume of Geometric Shapes Using Integration**
1. **Volume of the Right Circular Cylinder**
To find the volume of a right circular cylinder with a base of radius 4 in the xy-plane and height 5, evaluate the following integrals:
- **Option (a):**
\[
\int_0^5 \int_0^{2\pi} \int_0^4 dr\,d\theta\,dz
\]
- **Option (b):**
\[
\int_0^5 \int_0^{2\pi} r\,dr\,d\theta\,dz
\]
- **Option (c):**
\[
\int_0^5 \int_0^{2\pi} \int_0^2 dr\,d\theta\,dz
\]
2. **Volume Inside the Cone \(\phi = \pi/4\), \(0 \leq z \leq 4\)**
To find the volume of the region inside the cone where \(\phi = \pi/4\) and \(0 \leq z \leq 4\), evaluate these integrals:
- **Option (a):**
\[
\int_0^{2\pi} \int_0^{\pi/4} \int_0^{4\sqrt{2}} \rho\,d\rho\,d\phi\,d\theta
\]
- **Option (b):**
\[
\int_0^{2\pi} \int_0^{\pi/4} \int_0^4 \rho^2 \sin\phi\,d\rho\,d\phi\,d\theta
\]
- **Option (c):**
\[
\int_0^{2\pi} \int_0^{\pi/4} \int_0^{4\sec\theta} \rho^2 \sin\phi\,d\rho\,d\phi\,d\theta
\]
These integrals are used to calculate the volumes based on the specific geometrical constraints and limits provided in the integrals.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
