Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.4.72a

Chapter 6, Problem 6.4.72a

Explain the steps required to find the length of a curve x = g(y) between y=c and y=d.

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Identify the function given as \(x = g(y)\) and the interval over which you want to find the curve length, from \(y = c\) to \(y = d\).

Recall the formula for the length of a curve expressed as \(x = g(y)\), which is given by the integral: \[L = \int_{c}^{d} \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy\]

Compute the derivative \(\frac{dx}{dy}\) by differentiating \(g(y)\) with respect to \(y\).

Substitute \(\frac{dx}{dy}\) into the integral formula to get: \[L = \int_{c}^{d} \sqrt{1 + \left(g'(y)\right)^2} \, dy\]

Evaluate the integral over the interval \([c, d]\) to find the length of the curve.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Arc Length Formula for Parametric Curves

The arc length of a curve defined by x = g(y) between y = c and y = d is found by integrating the square root of 1 plus the derivative of x with respect to y squared. This formula accounts for the infinitesimal distances along the curve, summing them to find the total length.

Derivative of the Function x = g(y)

To apply the arc length formula, you need the derivative dx/dy, which measures how x changes with respect to y. This derivative is essential because it determines the slope of the curve and influences the length calculation by affecting the integrand.

Definite Integration over the Interval [c, d]

After setting up the integrand involving the derivative, you compute the definite integral from y = c to y = d. This integration sums the infinitesimal arc lengths along the curve, yielding the total length between the specified bounds.

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Definition of the Definite Integral

Related Practice

Textbook Question

Consider the following curves on the given intervals.

a. Write the integral that gives the area of the surface generated when the curve is revolved about the given axis.

y=tan x , for 0≤x≤π/4; about the x-axis

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Textbook Question

Emptying a cylindrical tank A cylindrical water tank has height 8 m and radius 2m (see figure).

a. If the tank is full of water, how much work is required to pump the water to the level of the top of the tank and out of the tank?

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Textbook Question

Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.

Theo: vT(t)=10, for t≥0

Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1

a. Graph the velocity function for both riders.

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Textbook Question

21–30. {Use of Tech} Arc length by calculator

a. Write and simplify the integral that gives the arc length of the following curves on the given interval.

y = cos 2x, for 0 ≤ x ≤ π

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Textbook Question

Region R is revolved about the line x=4 to form a solid of revolution.

a. What is the radius of a cross section of the solid at a point y in [1, 3]?