Find the area of the region between the curve y=3-x^2 and the line y=-1 by integrating with resp

Find the area of the region between the curve y=3-x^2 and...

Key Concepts

Area Between Curves

This concept involves finding the region enclosed by two or more curves. It is achieved by setting up an integral where one function is subtracted from the other, effectively measuring the vertical (or horizontal) distance between the curves over a given interval.

Definite Integration

Definite integration is the process of calculating the accumulation of quantities, such as area, over a specific interval. This technique is used to sum up infinitesimally small elements within the bounds of the region, yielding an exact area.

Method of Slicing (Integrating with Respect to a Variable)

When setting up an area integral, one must choose the appropriate variable of integration (x or y) based on the region's geometry. Vertical slices (integrating with respect to x) or horizontal slices (integrating with respect to y) are used depending on which orientation simplifies the description of the region and the formulation of the integral.

Determination of Intersection Points

Identifying where the curves intersect is crucial as these points serve as the limits of integration. They define the boundaries of the region whose area is being computed and ensure that the integral accurately reflects the extent of the area of interest.

Transcript

00:01 We want to find the area of the region between the curve y is equal to 3 minus x squared and the line y is equal to negative 1 by integrating with respect to both x and y.

00:12 So let's first go ahead and do x.

00:15 So that should give us.

00:21 So with respect to x.

00:26 So first the region we're interested in is this region right here.

00:30 So if we're integrating with respect to x, that means we want to do top to bottom or bottom to top so we start down here and we enter through at y is equal to negative 1 and then we exit through y is equal to 3 minus x squared so we'd want to integrate from well we would need to figure out what the bounds are in a moment but it would be the top function so 3 minus x squared minus our bottom function which would do you the same thing as just adding 1 d x now, we would want to set these two functions equal to each other to figure out what our balance of integration are.

01:16 So we'd have negative 1 is equal to 3 minus x squared, and you go through all the algebra, and we should get that x is equal to plus or minus 2.

01:26 All right.

01:30 And this actually ends up being an even function once we simplify it down.

01:37 So this is even.

01:38 So if you were to look at if f of negative x is equal to f of x, this would hold.

01:44 And the nice thing about even functions is we can rewrite this as two times integral from zero to the upper bound two.

01:50 And then let's simplify the end size, so it would just be four minus x squared dx.

01:55 And we want to do this because plug it in zero is easier than plug in negative two.

02:00 Now integrating, we'd have 4x minus 1 third x cubed, and we evaluate from 0 to 2.

02:11 So plugging in 2, we should get 8 minus 8 thirds, and then when we plug in 0, well, that should just give us 0.

02:25 Now, 8 minus 8 over 3 times 2.

02:30 That looks like it gives us 32 over 3.

02:38 So that would be our area of the region.

02:42 Now they tell us to also do this with respect to y.

02:47 So b is going to be with respect to y.

02:51 And unfortunately doing it this way, we'd have to actually work a little bit harder.

02:54 So we're going to need to solve for y.

02:59 So the blue or the purple line is already solved, so no issues there.

03:07 But our other equation, we're going to need to write this as...

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