Consider the area between the graphs x + 1y = 8 and x + 4 =...

Consider the area between the graphs x + 1y = 8 and x + 4 = y^2. This area can be computed in two different ways using integrals. First of all it can be computed as a sum of two integrals ?_a^b f(x) dx + ?_b^c g(x) dx where a = [ ], b = [ ], c = [ ] and f(x) = [ ] g(x) = [ ] Alternatively this area can be computed as a single integral ?_?^? h(y) dy where ? = [ ], ? = [ ] and h(y) = [ ] Either way we find that the area is [ ].

Transcript

00:01 Okay, so first of all let's sketch our region in the cartesian plan.

00:08 So this one, the first one is just a line with negative slope, which is y equals 8 minus x.

00:17 This one, okay, is just a parabola along the x -axis, and it is x equals y squared minus 4.

00:30 So let's draw our sketch here.

00:34 Let's sketch our region here.

00:37 Okay, so this is the x -axis, this is the y -axis.

00:42 Let's draw our parabola first.

00:44 This one is negative 4, and our parabola looks like this one.

00:51 Well, this one is negative 2.

00:55 This one is 2.

00:57 Now we got our line, y equals 8.

01:01 Minus x so our line okay at x equals 0 we got 8 here and at x equals 8 8 oh okay here we got x equals 8 so this is okay so our rego is exactly is exactly this one okay now here we can compute the area of this region by using integration with respect to x and with respect to y.

01:42 Now if you want to use integration with respect to x, we are going to have two integrals.

01:49 That is, the integral which is going to give me the area of this region, and the integral which is going to give me the area of this region.

02:00 So the first thing that we need to do is this, we are going to find this point of intersection.

02:06 Section here.

02:08 Okay, in order to find this point of intersection, we just need to solve the question, the quadratic equation, 8, 8 minus y squared plus 4 equals z.

02:22 Oh, okay, let me write it like this.

02:26 Okay, by using this one, we got x equals 8 minus y.

02:32 So we need to solve the equation 8 minus y equals y squared minus 4.

02:40 Okay so let's solve this quadratic equation.

02:46 Let's see if there is an easier way of doing this.

02:50 No i don't see it.

02:52 We just need to perform some computation...

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