뒤로Calculus Study Guide: Arc Length, Surface Area, Fluid Forces, and Center of Mass
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Q1. Find the length of the astroid defined by by calculating the length of half the first-quadrant portion, , for , and multiplying by 8.
Background
Topic: Arc Length of a Curve
This question tests your ability to use the arc length formula for a function to find the total length of a curve, specifically for a special curve called an astroid. You are asked to find the length of a segment in the first quadrant and use symmetry to get the total length.
Key Terms and Formulas
Arc Length Formula: For , :
Astroid: A curve defined by .
Symmetry: The astroid is symmetric about both axes, so you can compute a portion and multiply by the appropriate factor.
Step-by-Step Guidance
Express as a function of in the first quadrant: for .
Find the derivative of with respect to . Use the chain rule for differentiation.
Write the arc length integral for the specified interval :
Since the astroid is symmetric, multiply the result by 8 to get the total length.
Set up the integral, but do not evaluate it yet. Make sure your integrand is fully simplified and ready for calculation.
Try solving on your own before revealing the answer!
Final Answer:
The total length of the astroid is:
After differentiating and simplifying, the integral becomes:
Evaluating this integral gives the total length of the astroid. The answer is approximately $6$ (rounded to the nearest integer), but you should use a calculator or software to compute the exact value.
Q2. Find the lateral surface area of the cone generated by revolving the line segment , , about the y-axis. Check your answer with the geometry formula.
Background
Topic: Surface Area of a Solid of Revolution
This question tests your understanding of how to compute the surface area of a solid formed by revolving a curve about an axis, using calculus and comparing it to a geometric formula for a cone.
Key Terms and Formulas
Surface Area Formula (about y-axis):
Geometry formula for cone: , where is the base radius and is the slant height.
Step-by-Step Guidance
Identify the function and interval: , .
Compute for .
Set up the surface area integral for revolution about the y-axis:
Simplify the integrand and prepare to evaluate the definite integral.
Compare your result to the geometry formula for the lateral surface area of a cone.
Try solving on your own before revealing the answer!
Final Answer:
The calculus setup gives:
Evaluating the integral and comparing to the geometry formula (with , ) confirms the result.
Q3. A vertical right circular cylindrical tank measures 30 feet high and 20 feet in diameter. It is full of kerosene weighing 51.2 pounds per cubic foot. How much work does it take to pump the kerosene to the level of the tank?
Background
Topic: Work Done by Pumping Fluids
This question tests your ability to set up and evaluate an integral for the work required to pump a fluid to a certain height, using the concept of work as force times distance and integrating over the volume of the tank.
Key Terms and Formulas
Work: , where is the force at height .
Weight of fluid slab:
Distance to lift:
Step-by-Step Guidance
Let be the height above the tank bottom. The tank is 30 ft high, so .
Find the volume of a thin horizontal slab at height with thickness .
Compute the weight of this slab using the density of kerosene.
Determine the distance each slab must be lifted to the top of the tank.
Set up the integral for the total work required to pump all the kerosene to the top.
Try solving on your own before revealing the answer!
Final Answer:
The work required is:
Evaluating this gives ft-lb.
Q4. Calculate the fluid force on one side of the triangular plate submerged in water, using the coordinate system shown (pool surface at ).
Background
Topic: Fluid Force on Submerged Surfaces
This question tests your understanding of how to set up and evaluate an integral for the force exerted by a fluid on a submerged plate, using the concept of pressure varying with depth.
Key Terms and Formulas
Fluid Force:
Weight-density : The weight per unit volume of the fluid.
Depth: The vertical distance from the surface to the strip at .
Width function : The length of the plate at height .
Step-by-Step Guidance
Identify the limits of integration for (from the bottom to the top of the plate).
Express the width of the plate as a function of .
Write the depth of the strip below the surface as a function of .
Set up the integral for the fluid force using the formula above.
Simplify the integrand and prepare to evaluate the definite integral.
Try solving on your own before revealing the answer!
Final Answer:
The fluid force is:
Evaluating this gives (in appropriate units).
Q5. Find the centroid (center of mass) of a thin plate of constant density , bounded by the graphs of two functions and , for .
Background
Topic: Center of Mass (Centroid) of a Region Bounded by Two Curves
This question tests your ability to use integrals to find the centroid of a planar region of constant density, bounded above and below by two functions.
Key Terms and Formulas
Centroid :
Step-by-Step Guidance
Identify the functions and , and the interval .
Set up the integral for the mass of the region.
Set up the integral for the -coordinate of the centroid .
Set up the integral for the -coordinate of the centroid .
Prepare to evaluate the integrals and compute and .
Try solving on your own before revealing the answer!
Final Answer:
The centroid is:
Plug in the specific functions and interval to compute the numeric values.
