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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.2.65a
Chapter 6, Problem 6.2.65a

Determine whether the following statements are true and give an explanation or counterexample.


a. The area of the region bounded by y=x and x=y^2 can be found only by integrating with respect to x.

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First, identify the curves given: \( y = x \) and \( x = y^2 \). These two curves bound a region in the plane.
Next, consider the orientation of the curves. The curve \( y = x \) is a straight line, and \( x = y^2 \) is a parabola opening to the right.
To find the area bounded by these curves, you can integrate with respect to \( y \) because the parabola is expressed as \( x \) in terms of \( y \), and the line can be rewritten as \( x = y \). This makes it straightforward to set up the integral in terms of \( y \).
Alternatively, you can also express both curves as functions of \( x \) and integrate with respect to \( x \). For example, from \( y = x \), you have \( y = x \), and from \( x = y^2 \), you get \( y = \pm \sqrt{x} \). This allows you to set up the integral with respect to \( x \) as well.
Therefore, the statement that the area can be found only by integrating with respect to \( x \) is false, because the area can be found by integrating with respect to either \( x \) or \( y \), depending on which is more convenient.

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Key Concepts

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

Finding Area Between Curves

To find the area between two curves, you integrate the difference of their functions over the interval where they intersect. This can be done with respect to x or y, depending on which variable simplifies the integration and correctly represents the region.
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Finding Area Between Curves on a Given Interval

Integration with Respect to x vs. y

Integrating with respect to x involves expressing the top and bottom functions as y in terms of x, while integrating with respect to y involves expressing the right and left functions as x in terms of y. Choosing the variable of integration depends on the shape and boundaries of the region.
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Intersection Points and Region Boundaries

Determining the points where the curves intersect is essential to set the limits of integration. Understanding the region bounded by the curves helps decide whether integration with respect to x or y is more straightforward or even possible.
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Area of Polar Regions
Related Practice
Textbook Question

Consider a solid whose base is the region in the first quadrant bounded by the curve y=√3−x and the line x=2, and whose cross sections through the solid perpendicular to the x-axis are squares.


a. Find an expression for the area A(x) of a cross section of the solid at a point x in [0, 2].

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Textbook Question

Oil production An oil refinery produces oil at a variable rate given by Q'(t) = <1x3 matrix>, where is measured in days and is measured in barrels. 


a. How many barrels are produced in the first 35 days?

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Textbook Question

Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.


a. Suppose P=10, A=20, and r=0. If the initial population is N(0)=10, does the population ever become extinct? Explain.

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Textbook Question

Winding a chain A 30-m-long chain hangs vertically from a cylinder attached to a winch. Assume there is no friction in the system and the chain has a density of 5kg/m.

a. How much work is required to wind the entire chain onto the cylinder using the winch?

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Textbook Question

Find the area of the region (see figure) in two ways.

a. Using integration with respect to x.

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Textbook Question

Work in a gravitational field For large distances from the surface of Earth, the gravitational force is given by F(x) = GMm / (x+R)², where G = 6.7×10^−11 N m²/kg² is the gravitational constant, M = 6×10^24 kg is the mass of Earth, m is the mass of the object in the gravitational field, R = 6.378×10⁶ m is the radius of Earth, and x≥0 is the distance above the surface of Earth (in meters).


a. How much work is required to launch a rocket with a mass of 500 kg in a vertical flight path to a height of 2500 km (from Earth’s surface)?

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