Problem 44 Medium Difficulty

Graph the region between the curves and use your calculator to compute the area correct to five decimal places.

$ y = \cos x $ , $ y = x + 2 \sin^4 x $

Answer

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Video Transcript

we want to find the area of the region and closed by these two functions. The 1st 1 is in blue. That's Y equals Coast X, and the second function is in green. Why equals X plus to sign next to the fore? So we find by using a graphing calculator that there is actually two enclosed regions the 1st 1 here and the 2nd 1 over here. So we want to find the total area. So we're going to be adding these two regions area equals area one plus area two. So we're going to determine each of these areas separately. Area one is equal to what we're going to do. An integral. So I didn't say which is area when we're going to consider the the left most region to be area one. So we're going to be integrating from X equals negative. 1.912 two X equals negative 1.224 It's going to be our top function, minus our bottom function. The top function is the green one in this region. So this is X. You okay? Here we go. X plus two sign for X minus The bottom function, which is in Blue Coast X DX. So that's area one area to will be computed similarly so for area to which is this region the right region? Um, we're going to be into getting from our left most point which is now negative 1.224 to our right most point, which is 0.608 and we're going to do our top function minus their bottom function. This time I'm the functions play the opposite rules. The top function is the blue one now, which is co sex and we subtract the bottom function which is green so minus X minus to sign for X d x. We're allowed to use a into an integral calculator for this question. So after we compete these two areas and we add them together, we get her area. We are rounding to five decimal places, which means that our area that we obtain is 1.70 for 13 as the total area enclosed by both of these regions