Sketch the region enclosed by the given curves. Decide...

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to $ x $ and $ y $. Draw a typical approximating rectangle and label its height and width. Then find the area of the region.

$ y = \sin x $ , $ y = x $ , $ x = \frac{\pi}{2} $ , $ x = \pi $

Key Concepts

Sketching Regions

This involves drawing the graphs of the given functions to visualize the area of interest. A clear sketch helps in identifying the boundaries of the region that will be integrated and in choosing appropriate limits for the definite integrals.

Choosing the Order of Integration

Deciding whether to integrate with respect to x or y is based on the orientation and structure of the region. The best order minimizes the complexity of the integration limits, often making the integral easier to evaluate.

Area Between Curves

The concept involves calculating the area of a region bounded by two or more curves by setting up an integral of the difference of the functions. This requires determining the top and bottom (or right and left) curves over the interval of integration.

Approximating Rectangular Elements

In Riemann integration, the region is divided into small rectangles to estimate the area. The width and height of these rectangles are determined by the differential changes in x and y, respectively, which eventually leads to the formulation of the integral that represents the area.

Transcript

00:01 Okay, so we need to draw those given curves.

00:05 First, let's draw xxx, then we draw the first curve, which is sine x.

00:15 Okay, so let's draw sign x.

00:18 We start from negative pi, this will go to 0, and stop at pi.

00:26 Then we draw y equals to x, which is something like this.

00:32 It's like the tangent curve of sign x the origin and x equals to pi over 2 here okay x equals to pi over 2 this is when sign x equals 1 so here there will be 1 and this is x equals to pi so for the bounded region, it will be this shaded region right here.

01:26 So our job is evaluate this area.

01:35 So how do we do it? we find the approximating, since everything we can see here is represented by x.

01:43 So when we do the integral, we do it, with respect to x okay area will equal to the integral with respect to x so far so good and we need to find the boundary the boundary for x is just goes from um pi over 2 to pi and and this here is just the upper curve minus the lower curve um before that we can see uh we can draw a typical approximating rectangle.

02:26 Something like this, it's very small here.

02:31 If we zoom out, this will look like this.

02:38 Okay, this typical approximating rectangle, the valves will be delta x and height will be x xi minus sign.

03:00 Okay, so in other words, our integral will be x minus sine.

03:10 Okay.

03:12 Then we evaluate this integral to find area of the region...

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