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Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.

$ y = 1.3^x $ , $ y = 2 \sqrt{x} $

5.10615

Applications of Integration

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Oregon State University

Harvey Mudd College

University of Nottingham

Idaho State University

we want to find the area of the enclosed region by these two functions. The 1st 1 in blue is 1.3 to the X and the 2nd 1 in green is two times discredit of X. So using a graphing calculator, we find this enclosed region over here and we're going to determine this area. We also have the approximate points of intersection. Ah, so we're interested in these X values, so we're going to be integrating. So area is equal to integral from the lower X value 0.291 to the upper value 6.8 three and we're going to integrate our tough function minus their bottom function. Here, the top function is the green one. So two times a squared of X tooth, um, skirt of X and we're going to subtract the bottom function, which is in blue over here. So we subtract 1.3 to the exponents X dx on. We just want to calculate this. This will give us our approximation for the area. So let's just carry out the anti derivatives. The anti derivative of two times This crate of X is for over three x 23 over two. And the anti derivative of 1.3 to the ex is a 1.3 to the ex over lawn of 1.3 lung is d natural longer than and then we're going from 0.291 to 6.83 And then after we plug in these numbers, what we have is while plugging in 6.3 we get 1.20156 plus some more decimals and we subtract. Then we plug in 0.291 This is going to give us negative 3.90 for 59 Again, I cut some decimals s. So what we get as the approximate area is 5.1061 five.