Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or

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

Key Concepts

Setting Up the Area Integral

Once the appropriate variable of integration is chosen, the next step is to set up the definite integral that represents the area between the curves. This involves writing the integrand as the difference between the top (or rightmost) function and the bottom (or leftmost) function over the interval defined by the intersection points.

Typical Approximating Rectangle (Riemann Sum Concept)

In the process of integration, the area under a curve is approximated by summing the areas of numerous small rectangles. A typical approximating rectangle is drawn with a width corresponding to an infinitesimally small change in the variable of integration and a height representing the difference between the function values. This concept underlies the transition from Riemann sums to the exact area given by a definite integral.

Finding Intersection Points of Curves

Determining where curves intersect is crucial in establishing the limits of integration when computing areas. This involves setting the equations of the curves equal to each other and solving for the variable, which yields the boundaries of the enclosed region.

Graphing Functions and Regions

This concept involves understanding how to sketch the graphs of functions, identify their relative positions, and determine the area of the region they enclose. It requires recognizing the general shape and key features of different types of functions, such as radical functions and linear functions, and understanding how shifts and stretches affect these shapes.

Choosing the Variable of Integration

Deciding whether to integrate with respect to x or y is an important step in setting up the integral for area. The optimal choice simplifies the integration process by aligning the slices or rectangles (vertical or horizontal) with the geometry of the region, ensuring that either the height or the width of the slices corresponds directly to the difference between the functions.

Transcript

00:03 And this question we're going to be going over how to find the area between those two curves.

00:09 So what we'll do is we start with finding the intersection points.

00:14 I already did a little bit of this here, but what you do is you equate the two curves to each other, and then try and solve for x.

00:24 I did so by squaring both sides, and then just eventually getting a quadratic equation, which i very easily solved for x as shown here.

00:39 In order to figure out which is on top and which is on the bottom, what we need is we need to plug in a point within our integration interval and determine which has a greater value.

00:53 And when i do this, i found that if i put in x equals zero, i had the square root of 3, which is greater than the square root of 2, which is greater than 3 over 2.

01:07 That's pretty that's going to tell us that the square root is going to be the top function, and the linear function, it's actually a line, is on the bottom.

01:22 Now, remember that the area between two curves is given by the integrating over an interval of the top minus the bottom.

01:36 The whole thing integrated with respect to x.

01:38 And since i have what is on top and on the bottom and our integration bounds, i can plug in all of this information right now.

01:58 Remember that we found from one to three earlier.

02:04 Now, i can split this into two integrals because of the properties of the definite integral.

02:17 And i can, from the second integral, i can factor out a one -half...

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