123. Region between curves Find the area of the region bounded...

Question

123. Region between curves Find the area of the region bounded by the graphs of y = tan(x) and y = sec(x) on the interval [0, π/4].

Accepted Answer

Welcome back, everyone. In this problem, we want to find the area of the region bounded by the graphs Y equals sin X and Y equals cost X on the interval from 0 to 14th of pi. A says it's 1 square unit, B route 2 minus 1 square units. C route 2 + 1 square units, and D, root 2 square units. Now, let's try to visualize our graph here to help us figure out the area of our region. So here, if I draw a. Some access to the right. Right? And no 4 or axis or X X and Y axis, OK. We're looking for the area of the region bounded by the graph of Y equals sin X, which would look something like this, and Y equals cos X, which would look something like this, OK, on the interval from 0 to 1/4 of pi. OK. So here, essentially we're looking for this region that's shaded in red. OK, and let me just put the names of our functions here. So this would be Y equals X and down here it would be Y equals sin X, OK? Now how can we find the area of our region that's bounded here? What do we know? Well, recall, eh, that the area a is going to be equal to the definite integral. Of the top function minus the bottom function or the upper curve minus the lower curve all with respect to X. I know from our graph we can tell that Y equals cost X is the upper curve. Ye equals sin x is the lower curve, and our bones are from 0 to 4 of pi. So if we use all of that. Then this would imply that our area of our bonded region will be the definite integral between 0 and 14th of pi. Of Caus X. Minus sine x. All with respect the X. Now we can go ahead and integrate. The integral of class X is sin X. And the integral of negative sin X is positive cause X. And it will be between the bones of 0 and 4th of pi. So now we can apply our bones to help us solve. Now at X equals 14th of pi, then we'll have the sign of a 4th of pi. Plus the cosign of 1/4 of by. And that x equals 0 we'll have the sign of 0 plus the cosine of 0. No, when X equals 4th of pi, then the sign of a 4th of pi and the cost of 14th of pi are both root 2 divided by 2. So this is a half a fruit tool plus a half of fruit tool. On the other hand, the sign of 0 equals the cosine of 0. OK. Well, no, sorry, the sign of 0 is 0 and the cosine of 0 is 1, OK. So now, if we add both of these up. Route 2, sorry, a half of route 2 plus a half of route 2 equals route 2 and 0 + 1 equals 1. So our result is route 2 minus 1. Therefore, the area that's bounded by those curves would be equal to route 2 minus 1 square units. If we look back at our answer choices, that means B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

123. Region between curves Find the area of the region bounded by the graphs of y = tan(x) and y = sec(x) on the interval [0, π/4].

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