Sketch the region bounded by the graphs of the given equations, show a typical slice, approximat

Sketch the region bounded by the graphs of the given...

Key Concepts

Intersection of Curves

Finding the intersection points involves solving the system of equations of the given curves simultaneously. This step is crucial as the points of intersection determine the limits of integration and the boundaries of the region whose area is being calculated.

Graphical Representation of Regions

Sketching the curves provides a visual understanding of the region bounded by them. It helps identify which curve forms the left/right or upper/lower boundaries of the region, guiding the selection of the variable and the limits for integration.

Typical Slice (Cross-Section)

A typical slice, or cross-sectional element, is used to set up the integral for the area. By analyzing a small segment of the region, one can express the differential area (via a thin rectangle or strip) in terms of the integrating variable, which is then integrated over the entire region.

Setting Up Integrals

Once the boundaries and the typical slice are determined, the next step is to set up an integral that represents the area of the region. This involves selecting the correct limits of integration based on the intersection points and subtracting the functions appropriately to account for the region’s dimensions.

Definite Integration to Calculate Area

The area of the region is calculated by evaluating the definite integral that sums the contributions from all the thin slices across the bounded interval. This fundamental process of integration aggregates the differential areas into the total area of the region.

Estimation and Verification

Estimating the area, either by graphical analysis or numerical approximation, serves as a useful check against the calculated area from the integral. Verifying the result with an estimate helps ensure that the integral was set up correctly and that the calculation is reasonable.

Transcript

00:02 We have a bounded region that is represented with these two curves, and we are going to sketch it.

00:09 We're going to approximate the area, set up the integral to find the area, and then finally calculate that area.

00:16 Okay, so the first thing we do is we need to sketch these.

00:20 Now, with our first equation, x equals y squared and a little bit more, that is definitely a quadratic, but because it's x equals and it's in terms of y, that quadratic has to either face to the right or to the left.

00:35 So this is positive, so it's going to be open to the right.

00:40 So we'll be able to graph that one, and then our other equation is linear.

00:46 So let's go ahead and set our first equation equal to zero, so we know where it crosses the y -axis.

00:52 So it's going to cross the y -axis at x equals or sorry y -equals zero and at y -equals two.

01:01 And then as we talked about, it is going to face to the right.

01:07 Okay, so now our linear equation, we should go ahead and solve for x equals.

01:13 And of course, i know that it might be easier for you to solve a linear equation y -equals, but because our other function is in terms of x, we should also, make this function in terms of x so it's x equals y plus four so that four is your start value and you put that on your x axis and then it's a slope of one so we can see that we definitely have these two intersections but it's not obvious to where they are so the best thing to do to find these intersections is a set are two equations equal to each other so because they are um you know, the highest power is the power of two, and it's a quadratic, we are going to have to use our factoring methods.

01:58 So the best thing to do would be to move everything to one side.

02:02 So i'm going to go ahead and subtract the y and subtract the 4.

02:08 So now we can factor.

02:10 We need two numbers that multiply to negative 4, but add to negative 3.

02:15 So we will be using our 4 and 1, and then make sure that the 4 is negative, and so that you get that negative 3y for your middle.

02:23 Term.

02:25 So our two zeros are y equals negative one and y equals four.

02:30 So now we can go ahead and mark those on our graph.

02:33 And remember, we're going to be integrating in terms of y.

02:37 So when we do our integration start and stop, we need to make sure that they're y values.

02:44 Okay, so now we're on to right function minus left function.

02:48 Normally we would do top minus bottom, but because we're in terms of we look at our function that is mostly to the right.

02:55 So that's our linear function and then we subtract our quadratic.

03:01 Now when we actually go to do this subtraction, we've actually already subtracted the two functions when we moved everything over to the same side.

03:11 But notice we actually took our quadratic minus our linear...

Please give Ace some feedback