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Many of the geometric formulas that you've used in classes up to this point are can be derived through calculus. In this assignment, you will derive the general formula for the surface area of a sphere. In order to make sure the details are clear, be sure to follow the steps precisely, as the process is broken into steps along the way. So to frame what you're doing, the surface of a sphere can be through of as a rotational surface formed by taking a hemisphere and rotating it around an axis. Let's "center" the hemisphere over the origin, as pictured below. f(x) = Vr2 – x² x = -r x = r If we rotate the above graph around the x-axis, the result is a sphere. This is exactly what you'll do in this assignment. Note here that r is the unspecified radius of circle (an unspecified constant that will still be present in our resulting formula). Task: 1. Using the surface area integral form that we see in section 2.4, write the integral (with no simplification) that would give the surface area of the sphere. Be sure that your integral has the integral "S" symbol, the bounds of integration, the integrand, and the differential. 2. Simplify your integrand. I promise that it simplifies quite a bit. 3. Use the fundamental theorem of calculus to compute your integral. As a hint, your answer should have rin the expression, but not z by the time this step is done. 4. Now label your result as A ="what you got from step 3." Congratulations, you've derived the formula for the surface area of a sphere!

Many of the geometric formulas that you've used in classes up to this point are can be derived through calculus. In this assignment, you will derive the general formula for the surface area of a sphere. In order to make sure the details are clear, be sure to follow the steps precisely, as the process is broken into steps along the way. So to frame what you're doing, the surface of a sphere can be through of as a rotational surface formed by taking a hemisphere and rotating it around an axis. Let's "center" the hemisphere over the origin, as pictured below. f(x) = Vr2 – x² x = -r x = r If we rotate the above graph around the x-axis, the result is a sphere. This is exactly what you'll do in this assignment. Note here that r is the unspecified radius of circle (an unspecified constant that will still be present in our resulting formula). Task: 1. Using the surface area integral form that we see in section 2.4, write the integral (with no simplification) that would give the surface area of the sphere. Be sure that your integral has the integral "S" symbol, the bounds of integration, the integrand, and the differential. 2. Simplify your integrand. I promise that it simplifies quite a bit. 3. Use the fundamental theorem of calculus to compute your integral. As a hint, your answer should have rin the expression, but not z by the time this step is done. 4. Now label your result as A ="what you got from step 3." Congratulations, you've derived the formula for the surface area of a sphere!

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Many of the geometric formulas that you've used in classes up to this point are can be derived through calculus. In this assignment, you will derive the general formula for the surface area of a sphere. In order to make sure the details are clear, be sure to follow the steps precisely, as the process is broken into steps along the way. So to frame what you're doing, the surface of a sphere can be through of as a rotational surface formed by taking a hemisphere and rotating it around an axis. Let's "center" the hemisphere over the origin, as pictured below. f (x) = Vr2 – x² x = -r x = r If we rotate the above graph around the x-axis, the result is a sphere. This is exactly what you'll do in this assignment. Note here that r is the unspecified radius of the circle (an unspecified constant that will still be present in our resulting formula). Task: 1. Using the surface area integral form that we see in section 2.4, write the integral (with no simplification) that would give the surface area of the sphere. Be sure that your integral has the integral "S" symbol, the bounds of integration, the integrand, and the differential. 2. Simplify your integrand. I promise that it simplifies quite a bit. 3. Use the fundamental theorem of calculus to compute your integral. A a hint, your answer should have r in the expression, but not x by the time this step is done. 4. Now label your result as A ="what you got from step 3." Congratulations, you've derived the formula for the surface area of a sphere!
Transcribed Image Text:Many of the geometric formulas that you've used in classes up to this point are can be derived through calculus. In this assignment, you will derive the general formula for the surface area of a sphere. In order to make sure the details are clear, be sure to follow the steps precisely, as the process is broken into steps along the way. So to frame what you're doing, the surface of a sphere can be through of as a rotational surface formed by taking a hemisphere and rotating it around an axis. Let's "center" the hemisphere over the origin, as pictured below. f (x) = Vr2 – x² x = -r x = r If we rotate the above graph around the x-axis, the result is a sphere. This is exactly what you'll do in this assignment. Note here that r is the unspecified radius of the circle (an unspecified constant that will still be present in our resulting formula). Task: 1. Using the surface area integral form that we see in section 2.4, write the integral (with no simplification) that would give the surface area of the sphere. Be sure that your integral has the integral "S" symbol, the bounds of integration, the integrand, and the differential. 2. Simplify your integrand. I promise that it simplifies quite a bit. 3. Use the fundamental theorem of calculus to compute your integral. A a hint, your answer should have r in the expression, but not x by the time this step is done. 4. Now label your result as A ="what you got from step 3." Congratulations, you've derived the formula for the surface area of a sphere!
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