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.

검증된 단계별 안내

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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주요 개념

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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.

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.

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.

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.