![]() Lets see if we can imagine a three-dimensional shape whose base could be viewed as this shaded in region between the graphs of Y is equal to F of X and Y is equal G of X. Your bounds should obviously be the least and greatest x-values that lie on the circle. You should have the base length from the previous step, which is all you need to find the cross-sectional area.Ĥ. ![]() The cross-section is an equilateral triangle, and you probably learned how to calculate the area for one of those long ago. Remember that to express a circle in terms of a single variable, you need two functions (one for above the x-axis and one for below it, in this case).ģ. A width dx, then, should given you a cross-section with volume, and you can integrate dx and still be able to compute the area for the cross-section. You know the cross-section is perpendicular to the x-axis. Integrate along the axis using the relevant bounds.Ī couple of hints for this particular problem:ġ. Find an expression for the area of the cross-section in terms of the base and/or the variable of integration.Ĥ. Find an expression in terms of that variable for the width of the base at a given point along the axis.ģ. Figure out which axis (and thus which variable) you'll be using for integration.Ģ. So the surface area of this figure is 544, 544 square units.I won't give you the answer, but I'll offer a general strategy for questions of that variety:ġ. So one plus nine is 10, plus eight is 18, plus six is 24. Rotate it in our brains, although you could do that as well. To open it up into the net 'cause we could make sure We get the surface area for the entire figure, and it was super valuable And then you have thisīase that comes in at 168. The two magenta, I guess you could say, side panels, 140 plus 140, that's 280. Going to be, let's see, if you add this one and Which is equal to, let's see, 12 times 12 is 144, plusĪnother 24, so it's 168. ![]() Over here, which is this area, which is that area right over there. To figure out the area of, I guess you could say Of these, 14 times 10, they are 140 square units. It's also, this length right over here is the same as this length, so it's also 14 high and 10 wide. Now we can think about the areas of, I guess you could consider But that's also going toīe 48, 48 square units. If it was transparent, it would be this back Six times eight, which is equal to 48 whatever units, square units. Here is going to be 1/2 times the base, so times 12, It has a base of 12 and a height of eight. Of this right over here? Well, in the net thatĬorresponds to this area. So, what's, first ofĪll, the surface area? What's the surface area So the surface area of this figure, when we open it up, we can justįigure out the surface area of each of these regions. So if you were to open it up, it would open up into something like this, and when you open it up, it's much easier to figure out the surface area. You can't see it just now, it would open up into something like this. ![]() If you were to cut it right where I'm drawing this red,Īnd also right over here and right over there and right over there and also in the back where Made out of cardboard and if you were to cut it, It is if you had a figure like this, and if it was What's called nets, and one way to think about Surface areas of figures by opening them up into Want to do in this video is get some practice finding
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