Problem 44 - University Calculus: Early Transcendentals

Question

Sketch the region of integration and write an equivalent double integral with the order of integration reversed. $$\int_{0}^{\pi / 6} \int_{\sin x}^{1 / 2} x y^{2} d y d x$$

Step-by-Step Solution

Answer

The equivalent integral is \( \int_{0}^{1/2} \int_{\arcsin(y)}^{\pi/6} x y^2 \, dx \, dy \).

Step 1: Identify the region of integration

The limits of integration are given as \[ \int_{0}^{\pi/6} \int_{\sin x}^{1/2} x y^2 \, dy \, dx \] This means \( x \) varies from \( 0 \) to \( \pi/6 \) and for each \( x \), \( y \) varies from \( \sin x \) to \( 1/2 \).

Step 2: Sketch the region in the xy-plane

The region of integration in the xy-plane can be described:- For \( x \) between \( 0 \) and \( \pi/6 \),- \( y \) varies between \( \sin x \) and \( 1/2 \).This is a strip bounded below by the curve \( y = \sin x \) and above by \( y = 1/2 \), from \( x = 0 \) to \( x = \pi/6 \). Plot these bounds in the xy-plane to visualize the region.

Step 3: Determine the limits for reversing integration

To reverse the order of integration, first describe \( y \) bounds in terms of \( x \):- \( y \) ranges from \( 0 \) to \( 1/2 \).- For each fixed \( y \), \( x \) ranges from \( y = \sin x \) (i.e., \( x = \arcsin(y) \)) to \( x = \pi/6 \), based on the original limits.Confirm these intervals correctly cover the region identified in the sketch.

Step 4: Write the integral with reversed order

With the new limits of integration:\[ \int_{0}^{1/2} \int_{\arcsin(y)}^{\pi/6} x y^2 \, dx \, dy \]This new integral mirrors the original problem but reverses the order of integration.

Key Concepts

Region of Integration

The region of integration in a double integral is essential. It tells us exactly where we are integrating over the xy-plane. For the given integral \[ \int_{0}^{\pi/6} \int_{\sin x}^{1/2} xy^2 \: dy \: dx, \] the region can be imagined as a specific section in the plane that we need to calculate the area (or volume) over.

• The integral first indicates that \( x \) values range from 0 to \( \pi/6 \).
• For each \( x \) value, \( y \) takes on values between \( \sin x \) and 1/2.

To visualize this, think of it as tracing out a ribbon in the xy-plane. The lower boundary of the ribbon is defined by the curve \( y = \sin x \). The top boundary is a straight horizontal line where \( y = 1/2 \). Plot these lines to better understand the integration region. This plot shows us a small strip, where integration takes place.

Order of Integration

The order of integration refers to the sequence in which we integrate. In this exercise, we start by integrating with respect to \( y \) and then with respect to \( x \). This is due to the order given in the double integral notation, specifically the sequence of the limits when read from the inside out: \( dy \) followed by \( dx \).However, sometimes it’s more beneficial to switch this order, depending on the region's geometry or if one order simplifies the calculation. To reverse the order, we need to consider the limits differently:

• Start with \( y \), which now advances from 0 to 1/2, describing the possible vertical positions.
• Then, for each \( y \), determine \( x \) limits, from \( x = \arcsin(y) \) due to \( y = \sin x \), up to \( x = \pi/6 \).

By changing the order, we have the power to possibly make our calculations easier or to better address analytical constraints.

Limits of Integration

Limits of integration define the "edges" or "boundaries" for each variable we are integrating with respect to in a double integral. Correctly identifying these boundaries ensures we are capturing the right area or volume.In our original setup:

• \( x \) ranges from 0 to \( \pi/6 \)
• Within this \( x \) range, \( y \) spans from \( \sin x \) to \( 1/2 \)

However, when reversing the order, our perspective changes:

• \( y \) becomes the outer integral, moving between 0 and 1/2.
• For each fixed \( y \), \( x \) pushes its bounds from \( \arcsin(y) \), which equates to the original curve \( y = \sin x \), reaching up to \( \pi/6 \).

It's crucial to sketch or mentally visualize this shifting of limits to ensure accuracy and to comply with the actual region intended by the original integral setup.