The density of oil in a circular oil slick on the surface of...

The density of oil in a circular oil slick on the surface of the ocean at a distance of r meters from the center of the slick is given by ?(r) = 40 ? (1 + r²) kilograms per square meter. Find the exact value of the mass of the oil slick if the slick extends from r = 0 to r = 7 meters. Mass = Include units in your answer.

Transcript

0:00 Hello everyone.

00:01 In the problem we are given the density of oil in the slick in the form of s of r which is in the equation form of 40 divided by 1 added to r square kilogram per meter square.

00:14 Now in the question it has been asked if the exact value of the mass of the oil slick, if the slick extends from r is equal to 0 to r is equal to 7 meters.

00:26 Now we know that area of circular slick it is calculated as 2 pi r d r form.

00:32 In this case 2 pi r is the sur.

00:34 Circumference and there is thickness now we can integrate with respect to the given function and we'll get the total mass in the form of if we're taking total mass as capital m we can write it in the form of integration r is equal to 0 to r is equal to 7 because these are the limits the function be 40 divided by 100 to r squared multiplied by 2 pi r d r now we can simplify the integration in the form of m is equal to integration 0 to 7 2 pi and 40 these are constant number therefore we can take it outside of the integration we get 80 pi therefore m is equal to 80 pi multiplied by integration 0 to 7 r divided by 1 plus r square d r now we are going to solve the integration using substitution we are taking 1 plus r square is equal to t the differential form will give us 2 r d r d r is equal to d t which gives us r d r d r d r is equal to 1.

01:37 By 2 multiplied by d t we also have to change the limits therefore the limit was for r and it's going to change in t the r value was 0 to 7 putting these values we get 1 added to 0 that will give us 1 and putting 7 we get 7 square 49 added to 1 that will give us 50 now from 80 pi we are taking 2 for multiplication with r therefore we get 2 r d r we can put the value d t this will give us 45 limit being 1 to 50 instead of toward dr we are going to put the value d t divided by 1 plus r squared that will give us t...

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