Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. y=tan(11x) and y= sin(11x). -pi/33>= x<= pi/33
Then find the area S of the region. (Give an exact answer.)
To sketch the region enclosed by the given curves and decide whether to integrate with respect to x or y, we need to plot the graphs of the two equations: y = tan(11x) and y = sin(11x).
First, let's consider the interval -π/33 ≤ x ≤ π/33.
To plot the functions, we can use a graphing calculator or software. Here is a sketch of the region:
1.5|
| . . . . . . .
|. . . . . .
|... . . ...
| ... .
| .... .
0.5|____________.
|_______________.
0 |________________.
-1 0 1
| x
|
The blue curve represents y = tan(11x), and the red curve represents y = sin(11x). The region enclosed by the curves is between the two curves and between the x-values from -π/33 to π/33.
To find the area S of the region, we need to calculate the integral of the difference between the two curves over this interval. Since the curves intersect multiple times, we need to split the integral into several separate integrals.
Since the width of the approximating rectangles should be in the x-direction, we will integrate with respect to x.
Let's calculate the area for each section separately:
Section 1: From -π/33 to the first intersection point.
Section 2: From the first intersection point to the second intersection point.
Section 3: From the second intersection point to the third intersection point.
Section 4: From the third intersection point to the fourth intersection point.
Section 5: From the fourth intersection point to the fifth intersection point.
Section 6: From the fifth intersection point to π/33.
For each section, we need to calculate the difference between the upper and lower curves at each x-value and integrate that difference with respect to x.
Once we have the values for each section, we can add them together to find the total area S of the region.