Calculate the total area of the region bounded by the curve...

Name: Calculate the total area of the region bounded by the curve y = x^2 - 1, the x-axis, and the lines x = 0 and x = 4. Do not count area below the x-axis as negative. Round value to 4-decimal places.
Uploaded: 2022-08-09T01:16:27-08:00
Duration: 7 min 15 s
Channel: Kumareshwaran Rathinasabapathy
Description: Calculate the total area of the region bounded by the curve y = x^2 - 1, the x-axis, and the lines x = 0 and x = 4. Do not count area below the x-axis as negative. Round value to 4-decimal places.

Calculate the total area of the region bounded by the curve y = x^2 - 1, the x-axis, and the lines x = 0 and x = 4. Do not count area below the x-axis as negative. Round value to 4-decimal places.

Transcript

00:01 Let's find the total area of the region bounded by the curves y equals x squared minus 1, the lines x equal to 0, x equal to 4 and x -axis.

00:12 So first we draw a sketch of this curves and the lines and then identify the boundary region.

00:19 So when we do the sketch of these curves and the lines, we see that there are two bounded regions.

00:25 This one is one boundary region which i have shown in blue border.

00:29 And this is another boundary region which i have shown in red border so basically we have to calculate the area of these two regions and then add them up together so that we get the total area of the region bounded by these curves and the lines so now let's write the definite integral which is used to calculate the area under the curve y equals f of x so it is given by this definite integral area equals integral from x equals a to x equals b of y d x so this is the definite integral which gives the area under the curve y equals f of x between the lines x equals a and x equal to b and since here we have two regions we are going to write up a definite integral for this region and we also going to write a different integral for this region and then add them up together and particularly when we calculate the area of this region that is for this blue color region the curve y is below the x axis so when we calculate the definite integral we may come up with negative values and since it represents the area we count that as a positive area and so for that i'm going to write down this total area so this equals area of region 1 plus area of region 2 this blue colored 1 is basically region 1 this is region 1 the red colored 1 is this is region 2 so this equals i'm now writing the definite integral for to calculate the area of region 1 this is still under the curve y equals x squared minus 1 so first let's put the limits the limits all from 0 to 1 here x is 0 and at this point x is 1 so the limits will be from 0 to 1 of y d x and remember that since this area will come up as negative we can take its absolute value to count it as a positive area and then for region 2 this is under the curve y and the limits are from 1 to 4 so we let out this as 1 to 4 of y d x so let's replace the curve curve y equals x squared minus one in both the different integrals and then evaluate this total area and so this equals this one is 0 to 1 y is x squared minus 1 of d x we take its absolute value plus this one is from 1 to 4 y is x squared minus 1 d x so let's integrate this one integration of x squared minus 1 that will be x cube divided by 3 minus x.

03:48 We evaluate this from 0 to 1.

03:51 Remember that we also take the absolute value.

03:54 And we get the same curve.

03:57 So its integration will be x cubed divided by 3 minus integration of 1 is x...

Question

Calculate the total area of the region bounded by the curve y = x^2 - 1, the x-axis, and the lines x = 0 and x = 4. Do not count area below the x-axis as negative. Round value to 4-decimal places.

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