Ch. 6 - Applications of Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 6 - Applications of Integration

Problem 6.5.2

Chapter 6, Problem 6.5.2

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

Verified step by step guidance

Step 1: Recall the formula for the length of a curve. The length of a curve defined as x = g(y) between y = c and y = d is given by the integral:

\int_{c}^{d} \sqrt{1 + (\frac{d x}{d y}) 2} d y

Step 2: Compute the derivative dx/dy. To find the derivative of x with respect to y, differentiate the function x = g(y) with respect to y. This will give you

\frac{d x}{d y}

Step 3: Square the derivative dx/dy. Once you have computed

\frac{d x}{d y}

, square it to obtain

{(\frac{d x}{d y})}^{2}

Step 4: Substitute into the formula. Replace the squared derivative and the constant 1 into the square root part of the formula:

\sqrt{1 + {(\frac{d x}{d y})}^{2}}

Step 5: Evaluate the integral. Integrate the expression

\sqrt{1 + {(\frac{d x}{d y})}^{2}}

with respect to y from y = c to y = 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.

Parametric Representation of Curves

In calculus, curves can be represented parametrically, where one variable is expressed in terms of another. For the curve defined by x = g(y), y serves as the parameter. This representation allows us to analyze the curve's properties, such as its length, by integrating with respect to the parameter.

Arc Length Formula

The arc length of a curve defined by a function can be calculated using the formula L = ∫√(1 + (dx/dy)²) dy. Here, dx/dy is the derivative of x with respect to y, which measures the rate of change of x as y varies. This formula accounts for the infinitesimal changes in both x and y, providing the total length of the curve between specified limits.

Definite Integrals

Definite integrals are used to calculate the total accumulation of a quantity over an interval. In the context of finding the length of a curve, the limits of integration (c and d) define the segment of the curve being measured. Evaluating the definite integral gives the exact length of the curve between these two points, incorporating the contributions of all infinitesimal segments.

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Related Practice

Textbook Question

64–68. Shell method Use the shell method to find the volume of the following solids.

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Textbook Question

Lengths of symmetric curves Suppose a curve is described by y=f(x) on the interval [−b, b], where f′ is continuous on [−b, b]. Show that if f is odd or f is even, then the length of the curve y=f(x) from x=−b to x=b is twice the length of the curve from x=0 to x=b. Use a geometric argument and prove it using integration.

Textbook Question

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x = 2e^√2y + 1/16e^−√2y, for 0 ≤ y ≤ ln²/√2

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Textbook Question

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y = 2

Textbook Question

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis.

{Use of Tech} y = √50 -2x², in the first quadrant; about the x-axis

Textbook Question

y = -2