Area of regions Compute the area of the region bounded by the ...

Question

Area of regions Compute the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.

ƒ(𝓍) = 16―𝓍² on [―4, 4]

Accepted Answer

Hello. In this video, we are going to be computing the area of the region bounded by the graph of F of X is equal to 25 minus X2 and the X-axis over the interval from -5 to 5. So in order to approach this problem, let's just go ahead and graph the given information. First we are told that we have this. F of X function, which is a parabola facing the negative side. This parabola starts at 0.25 and is a parabola that opens downward with two X intercepts located at 5 and -5. Now, we want to find the area of the region that is bounded by this function and the X-axis, and we only want the area that is between -5 and 5. What that means is that we want to find the area. That is bounded between the two curves, which is the shaded region here. Now, in order to find this area, we need to set up a definite integral. Now, since we only want the area from -5 to 5, that means we only want to integrate on the region from -5 to 5, and we are going to be integrating about the function 25 minus X squared. So this is going to be the integral that gives us the area that we are looking for. Furthermore, notice that the area or the shape of the parabola is symmetric about the y axis. What this means is that in order to simplify our integral, we can only, we can get the same value of the original integral if we just double the area from 0 to 5, since this is a symmetric function. So, we can either get the same area by integrating from -5 to 5, or we can take twice the area from 0 to 5. Let's go ahead and integrate by using the second integral. Now, we will start by taking the derivative or the antiderivative of the given integral 25 minus X squared. That is going to give us the antiderivative 25 X minus 1/3 X cubed, and this is going to be evaluated from 0 to 5. Now, all we need to do is we need to plug in the upper bound into this into this expression. If we plug in the lower bound zero, it just zeros out the entirety of this expression. So we are left with 2 multiplied by 25, multiplied by 5 minus 1/3, multiplied by 5 cubed. This is going to simplify to be 2 multiplied by 125 minus 1/3 multiplied by 125. That will give us 125 divided by 3. Now, if we were to combine the two values together, that is going to give us Well, the distributing 2 1st is going to give us 250. Minus 250 divided by 3, and by combining these two terms together, we can rewrite 250 to be 750 divided by 3 minus 250 divided by 3, and this is going to simplify to be 500 divided by 3 square units. So that is going to be the area of the bounded region, and with that being said, the overall solution is going to be C. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Area of regions Compute the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.

ƒ(𝓍) = 16―𝓍² on [―4, 4]

Key Concepts

Definite Integral

Area Under a Curve

Sketching the Region

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