Represent the area of the given region by one or more...

Key Concepts

Sketching the Region

Sketching the curves and the region they enclose is a key step in understanding the geometry of the problem. A clear diagram helps in visualizing the area to be integrated, identifying the correct boundaries, and ensuring that the correct approach is taken when setting up the integral expressions.

Setting Up Integration Bounds

Setting up the appropriate bounds for integration involves deciding whether to integrate with respect to x or y based on the region's geometry. This concept requires an understanding of the relative positions of the curves and the corresponding limits for the variable of integration, which is essential for accurately defining the region's area.

Definite Integration

Definite integration is central to computing the area of a region bounded by curves. By integrating a function with proper limits, one can obtain the exact area under or between curves. In this context, knowing how to set up and evaluate these integrals is essential to representing the area of regions defined by intersections of curves.

Determining Intersection Points

Identifying the points where curves intersect is crucial when setting up integrals for areas. These intersection points serve as the limits of integration and help partition the region into parts that can be integrated separately. Recognizing and computing these intersections ensures that the integral accurately captures the entire area of the region.

Transcript

00:04 We want to represent the area of the region between these two curves by using a definite integral.

00:12 So let's take a look at the graphs of these two curves.

00:16 So they both represent circles, circles with different centers.

00:22 But the area, the region that we are speaking about is this area right in here.

00:32 Okay, the area between the function, so we're looking at this little kind of football -shaped, kind of almost oval -shaped region in here.

00:42 This is the area.

00:43 This is the region that we want to find the area of.

00:49 Now we have two choices.

00:51 We can integrate with respect to the x -axis, which wouldn't be too bad because the blue function stays above the green function.

01:01 You know, from x is zero to x is two.

01:06 And the same thing would actually happen if we integrated with respect to the y -axis, the blue function is staying above the green function.

01:15 So it doesn't matter which variable x or y we integrate with respect to.

01:22 So let's just integrate with respect to the x -axis.

01:25 And that means we have to rewrite both of these equations where we, y is written as a function of x.

01:34 So we have to take each of these equations, algebraically manipulate them, so y is written as a function of x.

01:46 All right, so if we start with this one over here, subtracting x squared from both sides, we get y squared equals 4 minus x squared, and then y would equal to positive or the negative.

02:04 We're actually only going to end up needing one of those.

02:07 But y would equal to positive or the negative square root of 4 minus x squared.

02:16 Let's take a look real quick.

02:20 Y equals the square root of 4 minus x squared.

02:29 Okay.

02:30 All right.

02:30 See it in purple? now it's blue.

02:33 Now it's purple.

02:34 Okay.

02:35 So you can see this is one of the functions that we need.

02:38 So we don't need the whole blue circle anymore.

02:40 So let's get rid of that.