Textbook Question
9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.
a. The displacement between t=0 and t=5
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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.
Verified step by step guidance1
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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03:39Integrals of Natural Exponential Functions (e^x)
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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07:45Area 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].
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?
Textbook Question
9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.
a. The displacement between t=0 and t=5
Textbook Question
A surface is generated by revolving the line f(x)=2−x, for 0≤x≤2, about the x-axis. Find the area of the resulting surface in the following ways.
a. Using calculus
Textbook Question
Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.
a. What is the radius of a cylindrical shell at a point x in [0, 2]?